A Wave is produced when a system is disturbed from its Equilibrium Position and this disturbance is traveling or propagating from one region to another. Water ripples, tidal waves, sound, earthquakes, wave in a string, radio and television transmission, visible light, infrared light, ultraviolet light, and x-rays are all examples of wave phenomena. In general, there are 3 types of waves: mechanical waves, electromagnetic waves, and matter waves. A Mechanical Wave is a wave that requires a medium in order to propagate. Water ripples, as well tidal waves, need water in order to travel from one region to another; without water, there can be no water ripples nor can there be tidal waves. Sound needs a medium as well as its motion involves the vibration of particles through variations in pressure. Water ripples, tidal waves, sound, earthquakes, and waves in a string are examples of mechanical waves. An Electromagnetic Wave is a wave that does not require a medium in order to propagate. Light from the Sun, for example, is able to reach the

Earth through empty space. Visible light, infrared light, ultraviolet light, x-rays, and radio and television transmission are all examples of electromagnetic waves. A Matter Wave is yet another type of wave phenomena. It pertains to the wavelike behavior of atomic and subatomic particles and forms a part of the foundation of quantum mechanics.

As a mechanical wave travels through a medium, particles of the medium undergo displacements from their equilibrium position. There are 2 types of mechanical waves according to the kind of displacement the medium undergoes: transverse waves and longitudinal waves. A Transverse Wave is a wave that causes displacements in the medium that are perpendicular or transverse to the direction of wave propagation.

v

A Longitudinal Wave is a wave that causes displacements in the medium that are parallel to the direction of wave propagation. Waves are generally periodic in nature. That is, particles of the medium undergo displacements that are similar to simple harmonic motion (SHM). Even seemingly complex waves can be considered periodic in that they can be shown as the sum of periodic waves. Waves, therefore, can be described in terms of the quantities found in simple harmonic motion. A = amplitude = maximum displacement from equilibrium that the medium can have f = frequency = cycles per unit time experienced by the medium = unit: s−1 = 1/s = 1 hertz = 1 Hz

T = period = amount of time for one cycle ω = angular frequency = frequency measured in radians per unit time k = wave number (spring constant in SHM) = unit: 1 rad/m In addition, a wave can also be described in terms of how long the wave is. λ = wavelength

v

speedwavek

fv

k

fT

f

===

=

=

=

ωλ

λππω

22

1

v y

x

λ

A

−A

Consider a periodic wave heading in the +x direction. The motion of the medium at x = 0 is given by the Wave Function:

After the wave has traveled some distance x(≠0), the displacement of the medium at x is the same as that of the displacement of the medium at x = 0 at the earlier time t − x/v. If the wave is heading in the −x direction, the wave function becomes:

EXAMPLE 1 The speed of sound in air at 20ºC is 344 m/s. If the frequency of a sound wave is 784 Hz, which corresponds to the note G5 on a piano, determine the period of the wave, as well as its wave number. How many radians will the wave have “traversed” in 15 seconds?

( ) ( ) ( )ftAtAtxy πω 2sinsin,0 ===

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −=

vxtfA

vxtAtxy

π

ω

2sin

sin,

( ) ⎟⎠⎞

⎜⎝⎛ −=⎟

⎠⎞

⎜⎝⎛ −=

λπωππ xtA

vfxftAtxy 2sin22sin,

( ) ( )kxtAtxy −= ωsin,

( )

( )kxtA vxtAtxy

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ +=

ω

ω

sin

sin,

( )

( )( ) radssHz

ft ft

mradssm

Hzvfk

fv fv

sHzf

T

stHzf

smv

25921.890,73157842

22

31981768.14344

784222

00127551.0784

1115784

344

==∆

=∆⇒=∆

=

====

=⇒=

===

===

πθ

πθπθω

ππλπ

λλ

T = 1.276 x 10−3 s

k = 14.320 rads/m ∆θ = 7.389 x 104 rads

Knowing the function for the position of particles in the medium as the wave passes through it, the velocity of the particles of the medium can be easily determined. It is important to note however that this velocity simply refers to the velocity of the particles as they move from and to the equilibrium position. In essence, the particles relatively remain in the same position as the wave does not take medium with when it moves. Similarly, the acceleration of the particles can also be determined.

It is interesting that, according to the above equations, the particles of the medium are exhibiting simple harmonic motion. This is yet another characteristic of a periodic mechanical wave! If the wave function is differentiated twice with respect to x: Using the equation for wave speed, the following equation is thus arrived at: This equation is known as the Wave Equation and has far-reaching implications. Any function that satisfies this equation is a periodic wave. It has been used extensively in quantum mechanics and, as such, has played a major role in the study of the structure of the atom.

( ) ( )( )( ) ( )kxtA

tkxtAv

kxtAtxytyv

y

y

−=∂

−∂=⇒

−=∂∂

=

ωωωω

cossinsin,

( )

( )( ) ( )

( )txya

kxtAt

kxtAa

ttxy

tv

a

y

y

yy

,

sincos

,

2

2

2

2

ω

ωωωω

−=⇒

−−=∂

−∂=⇒

∂∂

=∂

∂=

( ) ( ) ( )

( ) ( )2

2

2

2

2

2

222

2

,,

,sin,

ttxyk

xtxy

txykkxtAkx

txy

∂∂

=∂

∂⇒

−=−−=∂

∂

ω

ω

( ) ( )2

2

22

2 ,1,t

txyvx

txy∂

∂=

∂∂

A good example of a transverse wave is wave in a string. Wave in a string is produced by tying one of end of string, stretching it taut, and vibrating the free end of the string in SHM.

The free end of the string is vibrated by applying a transverse force Fy. The string gains a transverse momentum given by:

where: vx = longitudinal speed of the wave vy = transverse speed of the wave

As can be seen, Fy ∝ vy. Similarly, Fx ∝ vx since Fy and Fx can be considered as components of the resultant force while vy and vx can be considered as components of the wave’s overall velocity. By ratio and proportion:

where: v = speed of a wave in a string Fx = tension in the string µ = linear density of the string

EXAMPLE 2 One end of a 14.0-m-long rubber tube, with a total mass of 0.800 kg, is fastened to a fixed support. A cord attached to the other end passes over a pulley and supports an object with a mass of 7.5 kg. The tube is struck a transverse blow at one end. Find the time required for the pulse to reach the other end.

( ) yxyxyy

yfyfyiyfyy

tvvvtvmvtF

vmppppptF

µµ ===⇒

==−=−=∆= 0

Fx

L µ

µ

µ

µ

x2x

y

yx2x

yxx

yx

yxx

yxy

Fv

tvtvF

v

tvvv

tvF

tvvtvv

FtF

=⇒

=⇒

=⇒

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

µx

xFvv ==

x

yxy

x

y

x

y

vv

FF vv

FF

=⇒=

EXAMPLE 3 A transverse wave on a rope is given by:

y(x,t) = (0.750cm)sin{2π[(250s−1)t + (0.400cm−1)x]} a) Find the amplitude, period, wavelength, and speed of propagation. b) Is the wave traveling in the +x direction or −x direction? c) If the mass per unit length of the rope is 0.500 kg/m, find the tension in the rope.

( )( )

ssm

mvdt

vt Lby wave traveled distanced

sm mkg

NF speedwavev

NsmkggmWF

mkgmkg

Lm

kgmkgm

mL

rubber

x

objectobjectx

rubberrubber

object

rubber

390360029.086432768.35

14

86432768.35057142857.0

5.73

5.738.95.7

057142857.014

8.0

5.78.0

14

2

===⇒

===

=

===

====

===

==

=

µ

µ

t = 0.390 s

( ) ( )

( )( )

( )( ) smHzmfv

sHzf

1T

0.025m2.5cm k

Hzf f

cmradscmksradss

cmAx- towards moving

waveof function wavekxtAtxy

25.6250025.0

004.0250

1

2

2502

2

513274123.2400.02796327.570,12502

750.0

sin,

1

1

===

===

==⇒=

==⇒=

==

==

=

=+=

−

−

λ

λλπ

πωπω

π

πω

ω

A = 0.750 cm =0.00750 m

T = 0.004 s

λ = 2.5 cm =0.025 m

v = 6.25 m/s = 625 cm/s

The wave is traveling towards −x direction.

Consider a hollow pipe of cross-section area A filled with fluid of density ρ. The pipe is closed at one end and has a piston fitted in the open end. Longitudinal waves are produced inside the pipe when the piston is moved back and forth. The fluid is compressed/expanded and gains longitudinal momentum when the piston is moved back and forth.

A similar analysis can be done for longitudinal wave propagating through a solid rod or bar of cross-section A and density ρ. However, instead of Bulk Modulus, Young’s Modulus is used.

( )( ) NsmmkgvF

Fv Fv

mkg

x

xx

53125.1925.65.0

5.0

22

2

===

=⇒=

=

µ

µµ

µ

Fx =19.531 N

fluid F

fluid

fluidfluid

vtAvFt vtAVm

mvmvFt

ρρρ

=⇒==

=−=

0

0

ρρ

ρ

BtAv

BAtvv

vtAvv

vBAtFt

vv

BAF

vvAF

vtAtAvAF

VVAF

strainstress ModulusBulkB

fluid

fluid

fluidfluid

fluid

fluidfluid

==

==⇒

=⇒

==

∆===

2

0

ρBv =

ρYv =

EXAMPLE 4 a) In a liquid with density 1,300 kg/m3, longitudinal waves with frequency 400 Hz are found to have wavelength 8.00 m. Calculate the Bulk Modulus of the liquid. b) A metal bar with a length of 1.50 m has density 6,400 kg/m3. Longitudinal waves take 3.90 x 10−4 s to travel from one end of the bar to the other. What is Young’s Modulus for this metal? a) for the liquid:

b) for the metal bar:

EXAMPLE 5 What must be the stress in a stretched wire of a material whose Young’s Modulus is Y for the speed of longitudinal waves to equal 30 times the speed of transverse waves?

( )( )

( )( )Pa x

smmkgvB

Bv Bv

smmHzfvm

Hzfmkg

10

232

2

3

103312.1200,3300,1

200,384008400

300,1

=

==⇒

=⇒=

======

ρ

ρρ

λλ

ρ

B = 1.3312 x 1010 Pa

s x waveof travel of timetmkg6,400

by wave traveled distance mbar of lengthd

4

3

109.3

5.1

−==

=

===

ρ

( ) ( )Pax

mkgsmvY

Yv Yv

sms x

mtdv

10

322

2

4

104674556.9400,6153846.846,3

153846.846,3109.35.1

=

==

=⇒=

=== −

ρ

ρρ

Y = 9.4675 x 1010 Pa

( ) ( )LmF

VmY FY

FvYv

Fv

AFstress

translong

trans

900900

3030

=⇒=⇒

===

=

=

µρ

µρ

µ

When two waves overlap at a point at the same time, the resulting wave function at that point is the algebraic sum of the two waves’ wave functions for that point. The waves themselves are unaffected by the superposition at the overlap point; they continue propagating in their original directions, amplitudes, wavelengths, and frequencies. This is an important point to remember as it is easy to mistakenly think that when waves overlap, the original waves disappear to be replaced by a new wave. This is particularly true for two waves that overlap at all points at the same time. Due to superposition, it would appear that there is only one wave but in reality, there are two waves and the single wave that is being observed is merely the sum

of these two waves. The sum of the waves can be thought of as the overall effect of the two waves on the medium. For example, at a particular time t = t0 where: y1(x,t0), y2(x,t0) = individual waves at time t = t0 y(x,t0) = superposition of the two waves y1 and y2

( )Y x AF

YVFL

mFL

mYV

31011111111.1

900

900

−=⇒

=⇒

=

Stress = F/A = (1.1111 x 10−3)Y

( ) ( ) ( )txytxytxy ,,, 21 +=

y1(x,t0)

y1(x,t0)

y(x,t0)

Superposition of waves occurs when two waves overlap at the same point or region at the same time. The result of superposition of waves is called the Interference of the two waves. When the superposition of the waves at a point results in the total cancellation of the displacements of the medium caused by the waves, Destructive Interference of the waves is said to have occurred at that point. On the other hand, when the superposition of the waves at a point results in a larger displacement of the medium, Constructive Interference of the waves is said to have occurred at that point.

Put simply, Phase is the “angle” of the wave. When two identical waves but with a phase difference of 180º or π radians overlap at all points at the same time, the resulting interference is a wave

that appears to be stationary or “running in place.” This wave phenomenon is called a Standing Wave.

Nodes are points in the standing wave that remain in the same position as time passes. Midway between the nodes are the Antinodes. To emphasize the difference between standing waves and other types of waves, waves that do appear to be moving from one point to another are referred to as Traveling Waves.

Under the right conditions (i.e. by adjusting the length of the string, the tension in the string, etc.), waves in a string can lead to the formation of standing waves. Since the string is fixed at both ends, standing waves on the string must have nodes at both ends of the string.

( ) ( )( )tx,y of phase

AkxtAtxy=

=−=θ

θω sinsin,

y

x

N N N N N N

A A A A A

A = antinode N = node

Two Nodes:

Three Nodes:

Four Nodes:

Comparing them together, a pattern emerges:

If the length of string is kept constant, the above equation tells us that the wavelengths of the standing waves on the string are limited to certain values only: where: f1 = fundamental frequency = 1st harmonic frequency f2 = 1st overtone = 2nd harmonic frequency f3 = 2nd overtone = 3rd harmonic frequency n = harmonic number

2λ

λ

=

==

L

e wav standingof wavelength

stringof lengthL

λ=L

23λ

=L

⎟⎟⎠

⎞⎜⎜⎝

⎛ ===

segmentsor waves -half of numbern

nLK,3,2,1

2λ

µ

µ

λλ

λλ

xn

x

nnn

n

FnLf

Fv

nLfffv

nL

=⎟⎠⎞

⎜⎝⎛⇒

=

⎟⎠⎞

⎜⎝⎛===

==

2

2

2

( )

stringa in ves wa standingof sfrequencie harmonic

nfL

nvn FLnf x

n

=

==== 123,2,1

2K

µ

When a longitudinal wave travels through a column of fluid, the standing waves that can be produced under the right conditions depend upon the fluid container: a pipe open at both ends and a pipe closed at one end. I. Open Pipe

Comparing them together, a pattern emerges:

If L is kept constant, λ is limited to certain values.

II. Stopped Pipe

2λ

λ

=

==

L

wave standingthe of wavelengthpipe the of length L

λ=L2

3λ=L

( )K3,2,12

== n nL λ

ρ

ρ

λλ

λλ

BnLf

Bv

nLfffv

nL

n

nnn

n

=⎟⎠⎞

⎜⎝⎛⇒

=

⎟⎠⎞

⎜⎝⎛===

==

2

2

2

( ) 123,2,1

2nf

Lnvn B

Lnfn ==== K

ρ

4λ

λ

=

==

L

wave standingthe of wavelengthpipe the of length L

43λ

=L4

5λ=L

( )K7,5,3,14

== n nL λ

If L is kept constant, λ is limited to certain values.

EXAMPLE 6 A wire of mass 40.0 g is stretched so that its ends are tied down at points 80.0 cm apart. The wire vibrates in its fundamental mode with frequency 60.0 Hz and with an amplitude at the antinodes of 0.300 cm. a) What is the speed of propagation of transverse waves in the wire? b) Compute the tension in the wire.

EXAMPLE 7 An organ pipe has two successive harmonics with frequencies 1,372 Hz and 1,764 Hz. The speed of sound in air is 344 m/s. a) Is this an open pipe or a stopped pipe? b) What two harmonics are these? c) What is the length of the pipe?

( ) 147,5,3,1

4

4

nfL

nvn BL

nf

nL

n

n

===⎟⎠⎞

⎜⎝⎛=

==

Kρ

λλ

( ) ( )( )

smv

Hzm2Lfv Lvf

LnvF

Lnf

mcmAHzf

mcmLkggm

1

xn

96

608.0221

22

003.03.060

8.08004.040

1

1

=

==⇒=⇒

==

===

====

µ

( )( )

N m

smkgvLmF

Lmdensity linear

vF

Fv Fv

x

x

xx

8.4608.09604.0 2

2

2

2

=

=⎟⎠⎞

⎜⎝⎛=⇒

==

=⇒

=⇒=

µ

µ

µµ

v = 96 m/s

Fx =460.8 N

HzHzHzfsmvHzf

Hzf

n

n

392372,1764,1344

764,1372,1

1

=−=∆==

=

+

Since the harmonic number n should be a positive integer for the open pipe case, n = 3.5 is not allowed.

Resonance occurs when a system is vibrated with a periodically varying Driving Force at the same frequency as the Natural or Normal-Mode Frequency of the system. The system vibrates with a large amplitude or with an increasing amplitude as a result. Examples of resonance include building up the oscillations of a child on a swing by pushing with a frequency equal to the swing’s natural frequency, the vibrating rattle of a car when a certain engine speed is attained, the frequency tuner of a radio that allows one to select a particular radio station, and the uncontrolled swaying of a suspension bridge when a certain wind frequency strikes it. Resonance involving waves occur when a wave of frequency f travels through a system or medium with a normal-mode frequency f’ and f = f’. The natural frequency or normal-mode frequency of a system refers to the frequency at which the system will oscillate after being disturbed. For example, a simple pendulum will swing back and forth with a frequency dictated by the length of pendulum when it is released. When an object suspended via a spring is released, the object will oscillate up and down with a frequency dictated by

( )

( )( )( )

5.3344

372,143877551.0222

43877551.03922

3442

2222221

1

=

==⇒=

==∆

=

=−+=−+

=∆

−=∆ +

sm

HzmvLfn

Lnvf

mHz

smf

vL

Lv

Lnv

Lv

Lnv

Lnv

Lvnf

pipe, open Iffff

nn

nn

( )

( )( )

927

344372,143877551.044

4

43877551.042

442

=+=

==⇒=

=⇒=−+

=∆

n

smHzm

vLfn

Lnvf

mL Lv

Lnv

Lvnf

pipe, stoppeda For

nn

The pipe is a stopped pipe.

Harmonics: n = 7 & n = 9 L = 0.4388 m

the mass of the object and the spring constant of the spring. Some systems have multiple normal-mode frequencies as demonstrated by the harmonic frequencies of transverse waves passing through a stretched string, as well as the harmonic frequencies longitudinal waves passing through a column of air. Resonance in mechanical systems can be quite destructive. Bridges can be collapsed by a wind with the right frequency or even by soldiers marching over them with the right cadence. Buildings can be destroyed by relatively weak earthquake if the frequency of the earthquake coincides with those of the buildings. Glass can be shattered by a singer emitting a loud note with frequency corresponding exactly to one of the normal-modes of the glass.

Sound waves are longitudinal waves of pressure fluctuations that can travel through any gas, liquid, or solid. A periodic wave being the simplest type of mechanical wave, a simple sound wave is sinusoidal and has a definite frequency, amplitude, and

wavelength. The human ear is able to detect sound waves in the frequency range from about 20 Hz to 20,000 Hz. This is called the Audible Range of human hearing. Sound waves with frequencies greater than 20,000 Hz are called Ultrasonic Sound while those with frequencies lower than 20 Hz are called Infrasonic Sound. Sound waves typically travel in all directions from the source. They are capable of bending around corners, a phenomenon called Diffraction. Diffraction is the reason why people standing at the back of a speaker can still hear the voice of the speaker. Diffraction also explains why we can hear a police siren or an ambulance siren from around a corner. Like other waves, the speed of sound waves is more or less constant for a particular medium at a particular temperature. When the phase or temperature of the medium changes, the speed of sound waves passing through the medium also changes; this is particularly true for fluids. Likewise, when sound passes from one medium to another, it’s speed changes: vair(20ºC) = 344 m/s vhelium(20ºC) = 999 m/s vhydrogen(20ºC) = 1,330 m/s vliquid-helium(4K) = 211 m/s vmercury(20ºC) = 1,451 m/s vwater(0ºC) = 1,402 m/s

vwater(20ºC) = 1,482 m/s vwater(100ºC) = 1,543 m/s valuminum = 6,420 m/s vlead = 1,960 m/s vsteel = 5,941 m/s For temperatures near room temperature, the speed of sound in air for example is described by the following equation.

where: TC = temperature of air in Celsius scale

The physical characteristics of a sound wave such as amplitude and frequency are directly related to how it is qualitatively perceived by a listener. The greater the amplitude of the sound wave, the greater is the perceived Loudness. On the other hand, sound frequency is the primary factor in determining the Pitch; pitch is the quality of sound that allows us to classify the sound as being “high” or “low”. The greater the frequency of the sound wave, the higher the pitch that will be perceived by the listener.

Tones produced by different musical instruments may have the same fundamental frequency, which translates to having the same pitch, but they sound different became they contain different amounts of various harmonic frequencies (i.e. Harmonic Content). This difference in the tones is referred to as Tone Color, Quality, or Timbre. Timbre is often described using subjective terms like reedy, golden, round, mellow, and tinny. The Intensity of sound is defined as the time average rate at which energy is transported by the wave, per unit area. In other words, intensity is the average power of the wave per unit area (i.e. W/m2). The intensity of a sound wave is typically expressed in terms of Sound Intensity Level, β, which uses the Decibel Scale. The unit for β is decibel or dB. where: β = sound intensity level I = intensity of the wave I0 = reference intensity =1 x 10−12 W/m2

( )( ) smCTCsmv C 344206.0 +°−°⋅≅

( )0

log10IIdB=β

Here are some examples of sound intensity level of sound: βthreshold of hearing = 0 βrustle of leaves = 10 dB βaverage whisper = 20 dB βordinary conversation = 65 dB βbusy street traffic = 70 dB βriveter = 95 dB βthreshold of pain = 120 dB It is important to note that sound intensity varies with distance. The farther away the listener is from the source of the sound, the smaller the sound intensity detected by the listener. where: I1 = sound intensity heard by listener at distance r1 I2 = sound intensity heard by listener at distance r2 r1, r2 = distances from the source

It has been observed that the frequency of sound as perceived by a listener seemed to depend upon the motion of the sound source, as well as the motion of the listener even though the source is emitting sound

at constant frequency. That is, because of the motion of the sound source and the listener, there is a shift in the frequency of sound. This phenomenon is known as Doppler Effect. where: fL = frequency of sound heard by listener fS = frequency of sound emitted by source vL = speed of the listener = positive if heading in positive direction but negative if heading in negative direction vS = speed of the source = positive if heading in positive direction but negative if heading in negative direction v = speed of sound = positive if heading in positive direction but negative if heading in negative direction

21

22

2

1

rr

II=

SS

LL f

vvvvf

−−

=

vS

vL

fS fL

source listener