Learning Object 6

Standing Waves By David Park

Standing Waves

❖ A Standing wave is two harmonic waves with equal amplitude, wavelength and frequency that are moving in the opposite direction of each other.

❖ For mathematical simplicity, we assume that phase constant for both waves are zero.

❖ However, the general case is that the phase constant for the two waves may be unequal.

Standing Waves❖ The waves functions for the two opposing waves are: D1(x,t)=Asin(kx-wt)(For the wave moving in the direction of increasing x) D2(x,t)=Asin(kx+wt)(For the wave moving in the direction of decreasing x)

❖ When you apply the principal of superposition* the result is a wave function: D(x,t)=D1(x,t)+D2(x,t) =A sin(kx-wt)+A sin(kx+wt) =A[sin(kx-wt)+sin(kx+wt)]

❖ Using the trigonometric identity sin(a-b)+sin(a+b)=2sin(a)*cos(b), where a=kx and b=wtD(x,t)=2Asin(kx)*cos(wt)*(Superposition) When more than one wave is present in a medium at the same time, the resultant wave at any point in the medium is equal to the algebraic sum of the waves at that point

Travelling wave vs. Standing Wave❖ This equation may seem similar to a travelling wave,

however a travelling wave must have the position x and time t together in the form x∓vt, where v is the wave speed.

❖ In the equation D(x,t)=2Asin(kx)*cos(wt) the x and t variable are separate, with x in the sine function and t in the cosine. Differentiating the two

❖ D(x,t)=2Asin(kx)*cos(wt) with this equation, we can define a position-dependant amplitude A(x): A(x)=2A sin(kx)=2A sin(2π(x/λ))

❖ Using that equation, we can rewrite D(x,t)=2Asin(kx)*cos(wt) D(x,t)=A(x) cos(wt)

❖ When two waves of equal wavelength, frequency and amplitude but moving in opposite directions combine,each segment of the wave oscillates in a simple harmonic motion.

❖ The frequency and amplitude depends on the location of the segment along the wave

❖ All other points have amplitudes between zero and 2A and any two points that are one wavelength apart have the same amplitude because of the formula: A(xo+λ)=2A sin(2π((xo+λ)/λ)) =2A sin(2π(xo/λ)+2π) =2A sin(2π(xo/λ) = A(xo)

❖ The figure above shows a plot of A(x) as a function of position. Where the amplitude is a sine function, so certain points on the wave have zero amplitude and remain at rest at all times. These points are called nodes. The min/max points are called antinodes

Location of Nodes and Antinodes❖ At the nodes of a standing wave, A(x)=0 therefore, nodes occur when

sin((2π/λ)x)=0(2π/λ)x=mπ m=0, ±1, ±2,… x=m(λ/2) m=0, ±1, ±2,…

x=0, ±(λ/2), ±λ, ±(3λ/2), ±2λ,…

❖ Thus, the distance between two consecutive nodes is half a wavelength

❖ At the anti nodes of a standing wave, A(x) = ±2A, which occurs when

sin((2π/λ)x)= ±1(2π/λ)x=(m+(1/2))π m=0, ±1, ±2,… x=(m+(1/2))(λ/2) m=0, ±1, ±2,…

x=±(λ/4), ±(3λ/4), ±(5λ/4),…

❖ The distance between consecutive antinodes is also half a wavelength. An adjacent node and antinode are a quarter of a wavelength apart.

❖ To see how a wave oscillates in standing wave pattern we use the wave function in terms of the time period:

D(x,t)=2A sin((2π/λ)x)*cos((2π/T)t)

❖ All points between two consecutive nodes oscillate in phase with each other. The antinode has the greatest mean speed as it has to cover the longest distance (8A) in one period. The speed decreases the further away from the antinode and is zero at the nodes.

❖ Note: the motion of the sections between the next two nodes is π rad out of phase with the first section

❖ The sine term in the wave function accounts for this property:

A(xo+(λ/2))=2A sin((2π/λ)*(xo+(λ/2)))=2A sin((2π/λ)xo)+π=-2A sin((2π(λ/xo))

=-A(xo)

❖ The figure beside shows the displacement of a section of the oscillating string at intervals of T/8 from T=0 to t=T/2.

❖ The table below compares the displacements of the section of the wave between the first two nodes during the first half of a cycle. The motion for the next half of the period is in the opposite direction. This oscillatory motion repeats every cycle.

EXAMPLE

(Question) A rope is held tightly and shook until the standing wave pattern shown in the diagram at the right is established within the rope. The distance A in the diagram is 3.27 meters. The speed at which waves move along the rope is 2.62 m/s.a. Determine the frequency of the waves creating the standing wave pattern.

❖ Answer: 1.20Hz

❖ Since we’re given the velocity (v=2.62m/s) we can use the formula: v=f*λ and then rearrange it to find f. f=v/λ

❖ Looking at the graph of the wave we can see that there are 3/2 of a wave, so we equate that to the given distance, 3.27m = (3/2)λ, which equates to λ = 2.18m

❖ Plugging in the values back into f=v/λ, we get 2.62/2.18to get 1.20Hz

❖ Diagrams and tables all taken from Physics for Scientists and Engineers textbook.

❖ Example taken off of http://www.physicsclassroom.com/calcpad/waves/prob18.cfm