Sound and Oscillation

Periodic wave and its equation

The string in the figure above is taut, with point F being the source of periodic waves

In studies about periodic waves we saw that an oscillatory movement is nothing more than a periodic movement, that is, it is a movement that repeats itself continuously in an identical way. We have also seen that transverse waves are those whose propagation direction is perpendicular to the vibration direction, such as, for example, electromagnetic waves.

So, let’s consider the figure above, in which we have a stretched bungee cord. In the figure we can see the point F : this point on the string represents the source that produces the transverse periodic waves. We also have the point O, which represents the origin of the system xOy ; and P , which is a randomly chosen point on the string.

Let’s assume that initially t = 0 . Therefore, we can say that point F will perform a simple harmonic motion of amplitude A and initial phase θ 0 , so that the y -order of F will vary with time, following the MHS equation:

y=A.cos (ω.t+ θ 0 )

If we consider that there will be no loss of energy during the propagation of the wave along the string, during a time interval (Δt), we can say that the point P, located in the middle of the string, will also execute an MHS with the same amplitude A , but delayed Δt in relation to F.

Don’t stop now… There’s more after the publicity 😉

The time it took for the wave to reach point P is given by Δ t. Therefore, we have:

In the above equation the value of x corresponds to the value of the abscissa of P; and v is the wave propagation speed.

Therefore, we can say that the point P has the y-ordinate, given by the function of time:

y=A.cos[ω.(t-∆t)+θ 0 ]

As we know that ω = 2πf and that Δt = x/v, we have:

Substituting , follows:

For each point on the chord, the x -axis is fixed and the y -ordinate varies as a function of time, according to this function.

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