Question

What is Progressive wave? Derive an equation which represents progressive
wave.

Answer 1

Answer:

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Explanation:

Progressive Wave

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. Consider the wave to be 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 wavenumber 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.

Answer 2

A progressive wave is a type of wave that moves through a medium, such as a string or a fluid, with a constant speed and shape.

As it travels, the wave transfers energy from one point to another without any net displacement of the medium itself.

The mathematical equation that describes a one-dimensional progressive wave traveling in the positive x-direction can be written as:

y(x, t) = A sin(kx - ωt + φ)

where y is the displacement of the medium at a point x and time t,

A is the amplitude of the wave,

k is the wave number (related to the wavelength λ as k = 2π/λ),

ω is the angular frequency (related to the frequency f as ω = 2πf),

and φ is the phase constant (related to the initial phase of the wave).

The term kx - ωt + φ represents the phase of the wave,

which determines how the wave varies in space and time.

As time increases, the phase changes in a way that keeps the wave shape and speed constant, leading to the characteristic behavior of a progressive wave.

Note that the equation above assumes that the wave travels in a medium with a constant speed, which is often the case for simple wave systems. However, in more complex systems, the wave speed may vary with position or time, leading to more complicated wave behavior.

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