Question

what is a plane progressive simple harmonic wave ? establish equation for this wave in two different from​

Answer 1

Answer:

Explanation:

A wave which travels continuously in a medium in the same direction without the change in its amplitude is called a travelling wave or a progressive wave.

In this section, we will derive a function that will describe the propagation of a wave in a medium and gives the shape of the progressive wave at any instant of time during its propagation.

Let us consider the example of a progressive wave on a string. Here, we will describe the relation of displacement of any element on the string as a function of time and the vibration of the elements of the string along the length at a given instant of time.

Let y(x,t) be the displacement of an element at a position x and time t about the y-axis. Considering the wave as periodic and sinusoidal, the displacement of the element at a position x and time t, from the y-axis can be given as,

y (x, t ) = a sin (kx – ωt + φ ) …………………………………..(a)

We can write the above equation as a linear combination of sine and cosine function as,

y (x, t) =A sin (kx – ωt ) + B cos (kx – ωt ), …………(b) progressive wave

The equations (a) and (b) represent the transverse wave moving along the X-axis, where y(x,t) gives the displacement of the elements of the string at a position x at any time t, hence, the shape of the wave can be determined at any given time.

y(x, t) = a sin (kx + ωt + φ ),

The above equation represents a transverse wave moving along the negative direction of the X-axis.

Progressive Wave

The parameters that completely describe a harmonic wave are ‘a’, ‘φ’, ‘k’, and ‘ω’, where a is the amplitude, φ is the initial phase change, k is the angular wave number and ω is the angular frequency. Let us now learn in detail what these quantities represent.

Consider the sinusoidal graph shown above. Here, the plot shows a wave travelling in the positive X direction.

The point of maximum positive displacement is called a crest and that of maximum negative displacement is called a trough.

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