2.1 Transverse harmonic waves  (Page 9/6)

In this module, we shall consider wave motion resulting from harmonic vibrations. We shall discuss harmonic transverse wave in the context of a string. We assume that there is no loss of energy during transmission of wave along the string. This can be approximated when the string is light and taught. In such condition if we oscillate the free end in harmonic manner, then the vibrations in the string are simple harmonic motion perpendicular to the direction of wave motion. The amplitude of wave form remains intact through its passage along the string

Harmonic wave function

We know that a wave function representing motion in x-direction has the form :

$$y\left(x,t\right)=Af\left(ax-bt\right)$$

For the case of harmonic vibration, we represent harmonic wave motion in terms of either harmonic sine or cosine function :

$$y\left(x,t\right)=A\sin \left(kx-\omega t\right)$$

The argument of trigonometric function “kx-ωt” denotes the phase of the wave function. The phase identifies displacement (disturbance) at a particular position and time. For a particular displacement in y-direction, this quantity is a constant :

$$kx-\omega t=\text{constant}$$

where “k” is known as wave number. Its unit is radian/ meter so that the terms in the expression are consistent. Importantly, “k” here is not spring constant as used in the description of SHM.

Amplitude

The amplitude is the maximum displacement on either side of mean position and is a positive quantity. Since sine or cosine trigonometric functions are bounded between “-1” and “1”, the y-displacement is bounded between “-A” and “A”.

Harmonic oscillatory properties

Each particle (or a small segment of string) vibrates in simple harmonic motion. The particle attains the greatest speed at the mean position and reduces to zero at extreme positions. On the other hand, acceleration of the particle is greatest at extreme positions and zero at the mean position. The vibration of particle is represented by a harmonic sine or cosine function. For x=0 :

$$y\left(x=\mathrm{0,}t=t\right)=A\sin \left(-\omega t\right)=-A\sin \omega t$$

Clearly, the displacement in y-direction is described by the bounded sine or cosine function. The important point here is to realize that oscillatory attributes (like time period, angular and linear frequency) of wave motion is same as that of vibration of a particle in transverse direction. We know that time period in SHM is equal to time taken by the particle to complete one oscillation. It means that displacement of the particle from the mean position at a given position such as x=0 has same value after time period “T” :

$$y\left(x=\mathrm{0,}t=t\right)=y\left(x=\mathrm{0,}t=t+T\right)$$

$$\Rightarrow -A\sin \omega t=-A\sin \omega \left(t+T\right)=-A\sin \left(\omega t+\omega T\right)$$

We know that sine of an angle repeats after every “2π” angle. Therefore, the equality given above is valid when :

$$\Rightarrow \omega T=2\pi$$

$$\Rightarrow \omega =\frac{2\pi }{T}$$

The linear frequency is given by :

$$\Rightarrow \nu =\frac{1}{T}=\frac{\omega }{2\pi }$$

Wavelength and wave number

Wavelength is unique to the study of wave unlike frequency and time period, which is common to oscillatory motion of a particle of the string.

The wavelength is equal to linear distance between repetitions of transverse disturbance or phase. In terms of the speed of the wave, the wavelength of a wave is defined as the linear distance traveled by the wave in a period during which one cycle of transverse vibration is completed. We need to emphasize here that we can determine this wave length by choosing any point on the wave form. Particularly, it need not be a crest or trough which is otherwise a convenient reference point for measuring wavelength. For example, we can consider points A, B and C as shown in the figure. After a time period, T, points marked A’, B’ and C’ are at a linear distance of a wavelength, denoted by “λ”. Clearly, either of linear distances AA’ or BB’ or CC’ represents the wavelength.