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)
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.
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.
Amplitude is the magnitude of maximum displacement of a particle in a wave from the equilibrium position.
The image above shows the positive and negative amplitude in the case of a sinusoidal wave. But as we consider the magnitude of the displacement, the amplitude is always a positive quantity.
The argument (kx – ωt + φ) of the oscillatory term sin (kx – ωt + φ) is defined as the phase of the function. It describes the state of motion of the wave. Points on a wave which travel in the same direction, rising a falling together, are said to be in phase with each other. Points on a wave which travel in opposite directions to each other, such that, one is rising while the other is falling, are said to be in anti-phase with each other.
Wavelength (λ) is the distance between two identical points such as a crest or a trough on a wave parallel to the direction of propagation of the wave. It can also be defined as the distance over which the wave shape repeats itself. It is measured in metres (m).
Angular wave number
The wavenumber is the spatial frequency of a wave, in terms of cycles per unit distance. It can also be defined as the number of waves that exist over a specified distance, analogous to the concept of frequency.
Angular frequency is defined as the angular displacement per unit time of the rate of change of phase of a waveform.
Mathematically we can represent it as,
where T is the time period of the sinusoidal function representing the wave and f is the frequency.
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