Harmonic Waves LO 3_______________________________________________

Physics 101 LF2

By David Park

Harmonic Waves

• A wave generated by some source undergoing a simple harmonic motion is referred as a Harmonic wave, or a Sinusoidal wave

Harmonic Waves• You can imagine a harmonic

wave in the form of a string.

• One end of the string is attached to a source that oscillates (Yellow pendulum)

• This generates a continuous wave that travels along the string.

• Resulting in the creation of a harmonic motion (Figure shown beside)

Equations• At t=0 (Where the string

is attached to the source)The wave corresponds to A sine function (or cosine)

• This can be represented through the form: D(x)=Asin(kx) at t=0

• Where D(x) indicates the displacement of the string located at point x

• A is the Amplitude of the function

• k is the Wave Number

• Amplitude: The displacement D(x) oscillates between the point +/- A, the amplitude.

• The value of the amplitudedescribes the max. and min. displacement from the equilibrium

• The point of maximum displacement at D(x)=+A is called the crest and the point of minimum displacement (D(x)=-A) is called the trough.

• The graph above shows an example of amplitude for two different functions

• Before knowing how wave number (k) , relates to wave length (λ) it is good to know what a wave length is

• Wave length can be described as the shortest distance over which a wave repeats itself

• As shown in the figure above, one wave length extends from point x1 to x2, and two wave lengths extends from x1

to x2, repeating the shape of the crest twice

• The parameter k in the equation D(x)=Asin(kx) is called the Wave Number (Not to confuse with k describing spring constant, they are different)

• To describe the relationship between k and λ we refer to the fact that a waveform repeats over the length λ, therefore it can be represented as D(x)=D(x+λ) for any given location x

• Applying this relation to the equation, D(x)=Asin(kx) gives us a new equation,Asin(kx)=Asin(k(x+λ) which can be simplified to,Asin(kx)=Asin(kx+kλ)

Wave Number (cont.)

• We also know that the sine function repeats after 2π rad, so we can insert into a new equation to solve for k, kλ=2π radwhich can be rearranged to,k=2π/λ rad/m

• An example to show the relation between wave length and wave number -If λ=1m then k=2π and if λ=0.5m then k=4π

• This shows that as wave length increases (λ) , the wave number decreases (k)

Period and Wave frequency

• If the period of a source generating a continuous wave is T seconds, then one wave cycle is produced in T seconds

• The frequency (f) of a wave is equal to the number of wave cycles passing a fixed point of the medium in one second, therefore, in one second the medium will undergo f oscillations

Questions

Q1. In the figure shown below, which of the following wave comparisons are correct?

a)λ1<λ2<λ3 b)λ1>λ2>λ3c)λ1<λ3<λ2 d)λ1>λ2<λ3

• Answer: C, λ1<λ3<λ2Explanation: As stated before, one wave length is the shortest distance in which a wave repeats itself (crest to crest, trough to trough or midpoint to midpoint)and seeing from the figure in the question, λ1 has the shortest wavelength as it repeats more often than λ2 and λ3λ3 has a slightly smaller wave length than λ2, therefore, λ2 has the greatest wave length

Q2. Order the following waves from lowest to highest frequency

• Answer: b,a,c,d Explanation: Because b passes through the medium the least amount of times it has the lowest frequency. Same goes for the rest of the waves, and as d passes through the medium the most, it has the greatest frequency.