Physics Learning Object: Harmonic Waves

1. A Summary of Understanding Harmonic Waves!
Parameters of Harmonic Waves:!
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Using the simple graph of sin(x) above, we can identify it’s parameters:!
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The amplitude (A) is the maximum displacement from the graph’s position of
equilibrium, which in this and many cases, is zero. Therefore the maximum
displacement, and the amplitude of a general sine function is 1.!
! The crest ( ), is the maximum positive displacement and the trough ( )is
maximum displacement in the negative direction. !
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The wavelength (ƛ) is the smallest distance that a sinusoidal curve repeats itself. !
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The wave number (k) is a parameter of a harmonic wave equation and a quantity
related to the wavelength. It measures the change of phase per unit length. !
! k can be found by:!
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Combining this information of harmonic wave parameters, we get the wave function for
the transverse displacement of simple harmonic waves: !
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ƛ
A
D(x)= A sin(kx)

2. Speed, Velocity, and Acceleration!
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Wave Speed (v) is measured in respect to to the ‘frame of reference’, or in simpler
words, the one part of the wave you are watching. !
! Wave speed is given by the equation: !
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The displacement of travelling harmonic waves with a speed v, can be given by: !
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D(x,t)= A sin (k(x-vt)!
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by replacing x (from the previous equation) with x-vt. !
This equation can then be simplified by using relationships between k (wave number), w
(angular frequency), and the velocity, to give the relationship: kv=w , which simples the
equation to: !
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And here we have added the phase constant ɸ so that this equation becomes general
for all harmonic waves.!
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The general equation of transverse velocity and acceleration (instantaneous speed
and acceleration), are the first and second derivatives of the displacement function
above. !
Note~These equations have the same relationship as those of a simple harmonic
oscillator (but notice that for harmonic waves, our initial equation is sine not cosine!)!
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v= ƛf
D(x,t)= A sin (kx- wt + )
v(x,t)= -wA cos (kx- wt + )
a(x,t)= -w²A sin (kx- wt + )

3. Questions:!
1. A travelling harmonic wave is modelled by the function: D(x,t)= A sin (k(x-vt)), where
the amplitude is 0.4m, the wave number is 2.0rad/m, and the velocity is 3.0m/s.!
! a. What is period and frequency of the wave, assuming the wavelength is equal !
! to 2 pi?!
! b. What is the angular frequency of the function?!
! c. Assuming no phase constant, write an expression for the transverse velocity of
! the wave at time=t and position=x. !
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2. Using the following graph, determine which curve has the greatest and smallest wave
numbers. What is the relationship between wave number and frequency?!
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3. Using the following graph, identify which curve is the displacement, velocity, and
acceleration for a transverse harmonic wave. Then, describe which parameters of the
harmonic wave remain the same for d(t), v(t), and a(t), and which parameters change. !
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Answers!
A travelling harmonic wave is modelled by the function: D(x,t)= A sin (k(x-vt)), where the
amplitude is 0.4m, the wave number is 2.0rad/m, and the velocity is 3.0m/s.!
! a. What is period and frequency of the wave, assuming the wavelength is equal !
! to 2 pi?!
since v=ƛf, we can solve for f=(2pi)x(3.0m/s)=6pi≃18.8Hz!
since period is 1/f, T=0.053s!
! b. What is the angular frequency of the function?!
w=kv (see derivation in textbook) so w=2x3=6rad/s!
! c. Assuming no phase constant, write an expression for the transverse velocity of
! the wave at time=t and position=x. !
v(t)= -wA cos (kx - wt) where w=6, A= 0.4, and k=2!
v(t)= -2.4 cos (2.0x - 6t)!
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2. The yellow curve has the smallest k value, the blue, then the purple has the largest k
value. Wave number is directly proportional to frequency; as the wave number
increases, so does the frequency. !
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3. The black graph is displacement, the red is velocity, and the blue is acceleration. !
Amplitude increases, phase constant changes. Wavelength, wave number, and angular
frequency remain constant.