Example of a harmonic wave function

Problem statement:

A mechanical harmonic wave traveling along a rope can be expressed mathematically as (expressed in SI units):

  1. Find the amplitude, wavelength, period and speed of the wave.
  2. Find the maximum transverse speed and acceleration of an element of the rope.
  3. Draw a snapshot graph of the wave at t = 1/12 s.
  4. Find the equation of the simple harmonic motion undergone by the element of the rope located at x = 5 m from the origin.

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Solution:

The wave function of a harmonic wave is given by:

Identifying each term with what is given in the problem statement, the amplitude of the harmonic wave is:

The relationship between the wavenumber k and the wavelength λ is:

Therefore, isolating and substituting the numerical values we get:

The period T of a harmonic wave can be calculated from its angular frequency ω:

We can know the angular frequency from the wave function given in the problem statement. Then we isolate the period T:

The speed of the wave is given by:


The transverse speed of an element of the rope is calculated by taking the first derivative of the wave function y(x,t) with respect to time:

Therefore, the maximum transverse speed is:

The acceleration of an element of the rope is the derivative of its speed with respect to time:

And its maximum value:

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In order to draw a snapshot graph of the wave at a certain time, we have to substitute that time into the wave function y(x,t). This way we will obtain a function y(x) that is a snapshot of the rope at that time.

If we substitute t = 1/12 s into the wave function we get:

And plotting the function on a cartesian plane:


When a harmonic wave travels along a medium, each element of the medium undergoes simple harmonic motion. In order to find the equation y(t) of this motion we substitute the value of its x coordinate into the wave function.

Substituting x = 5 m into the wave function we get:

The expression above describes a simple harmonic motion. Plotting it on a Cartesian plane we get:

As you can see, the variable plotted on the horizontal axis is the time, because we are seeing the motion of just one element of the rope (x is fixed).

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