At any time t, the displacement y of the element located at position x is given by
`y (x, t ) = a sin (kx - ωt + phi )....(1)`
One can as well choose a cosine function or a linear combination of sine and cosine functions such as,
`y (x, t) =A sin (kx - ωt ) + B cos (kx - ωt )....(2)`
then in equation (1)
`a=sqrt(A^2+B^2)`
and `phi=tan^(-1)(B/A)`
Functions, such as that given in Eq. (1), represent a progressive wave travelling along the positive direction of the x-axis. On the other hand a function,
`y(x, t) = a sin (kx + ωt + phi )` represents a wave travelling in the negative direction of x-axis.
A point of maximum positive displacement in a wave, shown by the arrow, is called crest, and a point of maximum negative displacement is called trough.
The progress of the wave is indicated by the progress of the short arrow pointing to a crest of the wave towards the right. As we move from one plot to another, the short arrow moves to the right with the wave shape, but the string moves only parallel to y-axis.
It can be seen that as we go from plot (a) to (e), a particular element of the string has undergone one complete cycle of changes or completed one full oscillation.
`text(Amplitude and Phase :)`
The `text(amplitude)` `a` of a wave is the magnitude of the maximum displacement of the elements from their equilibrium positions as the wave passes through them. It is depicted in Fig. (a). Since `a` is a magnitude, it is a positive quantity, even if the displacement is negative.
The `text(phase)` of the wave is the argument `(kx - ωt + phi)` of the oscillatory term sin `(kx - ωt + phi)` in Eq. (1).
It describes the state of motion as the wave sweeps through a string element at a particular position x.
It changes linearly with time t.
The sine function also changes with time, oscillating between +1 and -1. Its extreme positive value +1 corresponds to a peak of the wave moving through the element; then the value of y at position x is a.
Its extreme negative value -1 corresponds to a valley of the wave moving through the element, then the value of y at position x is -a. Thus, the sine function and the time dependent phase of a wave correspond to the oscillation of a string element, and the amplitude of the wave determines the extremes of the elements displacement.
The constant `phi` is called the `text(initial phase angle)`.
`text(Wavelength and Angular Wave Number :)`
The wavelength λ of a wave is the distance (parallel to the direction of wave propagation) between the consecutive repetitions of the shape of the wave. It is the minimum distance between two consecutive troughs or crests or two consecutive points in the same phase of wave motion.
A typical wavelength is marked in Fig. (a), which is a plot of Eq. (1) for `t = 0` and `phi = 0`. At this time Eq. (1) reduces to
`y(x, 0) = a sin kx`
By definition, the displacement y is same at both ends of this wavelength, that is at `x = x_1` and at `x = x_1 + λ`. Thus, by Eq. (1)
`a sin k x_1 = a sin k (x_1 + λ)`
`= a sin (k x_1 + k λ)`
This condition can be satisfied only when,
`k λ = 2πn`
where n = 1, 2, 3... Since `λ` is defined as the least distance between points with the same phase,
n =1 and
`k=(2pi)/lamda`
k is called the `text(propagation constant)` or the `text(angular wave number)`; its SI unit is radian per metre.
`text(Period, Angular Frequency and Frequency :)`
From equation (1)
At `x=0`
`y (0,t) = a sin (-ωt)`
`= -a sin ωt`
The period of oscillation T of a wave is defined as the time any string element takes to move through one complete oscillation.
Applying Eq. (1) on both ends of this time interval, we get
`- a sin ωt_1 = - a sin ω(t_1 + T )`
`= - a sin (ωt_1 + ωT )`
This can be true only if the least value of `ωT` is `2π`, or if
`ω = 2π/T`
`ω` is called the angular frequency of the wave, its SI unit is rad`s^(-1)`.
The frequency `nu` of a wave is defined as `1//T` and is related to the angular frequency `ω` by
`nu=1/T=omega/(2pi)`
It is the number of oscillations per unit time made by a string element as the wave passes through it. It is usually measured in hertz.
At any time t, the displacement y of the element located at position x is given by
`y (x, t ) = a sin (kx - ωt + phi )....(1)`
One can as well choose a cosine function or a linear combination of sine and cosine functions such as,
`y (x, t) =A sin (kx - ωt ) + B cos (kx - ωt )....(2)`
then in equation (1)
`a=sqrt(A^2+B^2)`
and `phi=tan^(-1)(B/A)`
Functions, such as that given in Eq. (1), represent a progressive wave travelling along the positive direction of the x-axis. On the other hand a function,
`y(x, t) = a sin (kx + ωt + phi )` represents a wave travelling in the negative direction of x-axis.
A point of maximum positive displacement in a wave, shown by the arrow, is called crest, and a point of maximum negative displacement is called trough.
The progress of the wave is indicated by the progress of the short arrow pointing to a crest of the wave towards the right. As we move from one plot to another, the short arrow moves to the right with the wave shape, but the string moves only parallel to y-axis.
It can be seen that as we go from plot (a) to (e), a particular element of the string has undergone one complete cycle of changes or completed one full oscillation.
`text(Amplitude and Phase :)`
The `text(amplitude)` `a` of a wave is the magnitude of the maximum displacement of the elements from their equilibrium positions as the wave passes through them. It is depicted in Fig. (a). Since `a` is a magnitude, it is a positive quantity, even if the displacement is negative.
The `text(phase)` of the wave is the argument `(kx - ωt + phi)` of the oscillatory term sin `(kx - ωt + phi)` in Eq. (1).
It describes the state of motion as the wave sweeps through a string element at a particular position x.
It changes linearly with time t.
The sine function also changes with time, oscillating between +1 and -1. Its extreme positive value +1 corresponds to a peak of the wave moving through the element; then the value of y at position x is a.
Its extreme negative value -1 corresponds to a valley of the wave moving through the element, then the value of y at position x is -a. Thus, the sine function and the time dependent phase of a wave correspond to the oscillation of a string element, and the amplitude of the wave determines the extremes of the elements displacement.
The constant `phi` is called the `text(initial phase angle)`.
`text(Wavelength and Angular Wave Number :)`
The wavelength λ of a wave is the distance (parallel to the direction of wave propagation) between the consecutive repetitions of the shape of the wave. It is the minimum distance between two consecutive troughs or crests or two consecutive points in the same phase of wave motion.
A typical wavelength is marked in Fig. (a), which is a plot of Eq. (1) for `t = 0` and `phi = 0`. At this time Eq. (1) reduces to
`y(x, 0) = a sin kx`
By definition, the displacement y is same at both ends of this wavelength, that is at `x = x_1` and at `x = x_1 + λ`. Thus, by Eq. (1)
`a sin k x_1 = a sin k (x_1 + λ)`
`= a sin (k x_1 + k λ)`
This condition can be satisfied only when,
`k λ = 2πn`
where n = 1, 2, 3... Since `λ` is defined as the least distance between points with the same phase,
n =1 and
`k=(2pi)/lamda`
k is called the `text(propagation constant)` or the `text(angular wave number)`; its SI unit is radian per metre.
`text(Period, Angular Frequency and Frequency :)`
From equation (1)
At `x=0`
`y (0,t) = a sin (-ωt)`
`= -a sin ωt`
The period of oscillation T of a wave is defined as the time any string element takes to move through one complete oscillation.
Applying Eq. (1) on both ends of this time interval, we get
`- a sin ωt_1 = - a sin ω(t_1 + T )`
`= - a sin (ωt_1 + ωT )`
This can be true only if the least value of `ωT` is `2π`, or if
`ω = 2π/T`
`ω` is called the angular frequency of the wave, its SI unit is rad`s^(-1)`.
The frequency `nu` of a wave is defined as `1//T` and is related to the angular frequency `ω` by
`nu=1/T=omega/(2pi)`
It is the number of oscillations per unit time made by a string element as the wave passes through it. It is usually measured in hertz.