This document discusses harmonic waves and travelling harmonic waves. It provides relevant equations such as the wave number k being equal to 2π divided by the wavelength λ. The angular frequency ω is equal to 2π divided by the period T. The displacement of a travelling harmonic wave D(x,t) is described by the equation D(x,t) = A sin(kx - ωt) where A is the amplitude, k is the wave number, x is position, ω is angular frequency, and t is time. An example problem is worked through to find the displacement, wavelength, and frequency of a travelling harmonic wave given its displacement equation.

1. Travelling
Harmonic
Waves
By Yungee Jung

2. General Motions of Waves
• Waves are a motion of disturbance
• A wave that undergoes simple harmonic
motion is called a Harmonic Wave.
As they oscillate, it generates a continuous wave
that travels along the string in a simple harmonic
motion.
• There are travelling Harmonic Waves; its
displacement from its equilibrium position
changes with time as wave passes through it.

3. Relevant equations:
• Harmonic waves have a wave number: denoted as
“k”: this is a parameter. It measures the change of
phase per unit length of the string that is in
harmonic motion.
• There is a relationship between K and the
wavelength (λ ):
Since the sin function repeats after 2π rads,
K= 2π/λ : The waveform repeats over the length of λ;
the sin function repeats after 2π. Therefore, the wave
number is in radians/ meter.

4. Relevant equations
• The wave number “k” is the measure of change of
phase per unit length.
• Relation with λ : As wavelength increases, wave
number decreases, as shown in K= 2π/λ.
Like any other Harmonic motions, the angular
frequency: “w” = 2π/T.
• One wave cycle is produced in T seconds. This is
called the period “T.”

5. Equation of Travelling
Harmonic Wave
• It is similar to describing a moving pulse.
The displacement of an element on the string of the
wave is denoted by:
• D(x,t) = A sin(kx-wt). The wave is travelling in
direction of increasing x. ( The wave moves to the
right
• “A” symbolizes amplitude: the wave oscillates
between values + A and –A.
• D(x,t) = A sin(kx-wt) shows the wave travelling in
direction of decreasing x. (The wave moves to the
left)

6. Question of travelling
Harmonic Wave:
A harmonic wave travels along a string, described by
the wave function below:
D(x,t) = (0.50m) sin[(4.0 rad/m)x – (0.5 rad/s)t]
The x is measured in meters, and t is in seconds
a) what is the displacement of a segment of string at
x= 0.20m at t=1.0 seconds?
b) Find the wavelength and frequency of the wave.

7. Solution to part a
Finding the displacement at x= 0.20 m, and t= 1.0 sec:
substitute in the values for x and t to get the D( x,t)
value:
D(0.20, 1.0) = (0.50) sin[(4 x 0.20) – 0.5(1.0)]
= 0.148 m of displacement.

8. Solution to Part b
• First, the equation can be re written in terms of λ: (
by substituting equations):
D(x, t) = 0.50 sin(2π/λ)x – (2π/T)t Since k = 2π/λ and w
= 2π/T.
Therefore, 4.0 = 2π/λ and λ= 2π/4 which is
approximately 1.57m
Next, in order to solve for frequency, we must first solve
for the period, T.
T = 2π/w. So, T= 2π/0.5, which is 4π. Since frequency is
1/T, the frequency is equal to 1/4π.

9. Solutions
Overall, the frequency calculated was 1/4π and the
wavelength calculated was 1.57meters.
Once an equation is constructed, you can easily
calculate the period, angular frequency, and the
wavelength.
The Key equations used were : k = 2π/λ and The
position/ displacement D(x,t) sin function from the
equilibrium position.