A mechanical harmonic wave traveling along a rope can be expressed mathematically as (expressed in SI units):
- Find the amplitude, wavelength, period and speed of the wave.
- Find the maximum transverse speed and acceleration of an element of the rope.
- Draw a snapshot graph of the wave at t = 1/12 s.
- 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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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).The post Example of a harmonic wave function appeared first on YouPhysics