Standing Waves and Normal Modes
`color{blue} ✍️` We considered above reflection at one boundary. But there are familiar situations (a string fixed at either end or an air column in a pipe with either end closed) in which reflection takes place at two or more boundaries.
`color{blue} ✍️` In a string, for example, a wave going to the right will get reflected at one end, which in turn will travel and get reflected from the other end. This will go on until there is a steady wave pattern set up on the string. Such wave patterns are called `"standing waves"` or stationary waves.
`color{blue} ✍️`To see this mathematically, consider a wave travelling along the positive direction of x-axis and a reflected wave of the same amplitude and wavelength in the negative direction of x-axis. From Eqs. (15.2) and (15.4), with φ = 0, we get:
`color{blue} ✍️`The resultant wave on the string is, according to the principle of superposition:
`color{blue} ✍️`Using the familiar trignometric identity
`color{purple} {Sin (A+B) + Sin (A–B) = 2 sin A cosB}` we get,
`color{brown} {"Note"}` the important difference in the wave pattern described by Eq. (15.37) from that described by Eq. (15.2) or Eq. (15.4).
`color{blue} ✍️`The terms `kx` and `ωt` appear separately, not in the combination `kx - ωt.` The amplitude of this wave is `2a sin kx.`
`color{blue} ✍️`Thus in this wave pattern, the amplitude varies from point to point, but each element of the string oscillates with the same angular frequency `ω` or time period. There is no phase difference between oscillations of different elements of the wave.
`color{blue} ✍️`The string as a whole vibrates in phase with differing amplitudes at different points. The wave pattern is neither moving to the right nor to the left. Hence they are called `"standing or stationary waves."`
`color{blue} ✍️`The amplitude is fixed at a given location but, as remarked earlier, it is different at different locations. The points at which the amplitude is zero (i.e., where there is no motion at all) are nodes; the points at which the amplitude is the largest are called `"antinodes."`
`color{blue} ✍️`Fig. 15.12 shows a stationary wave pattern resulting from superposition of two travelling waves in opposite directions.
`color{blue} ✍️`The most significant feature of stationary waves is that the boundary conditions constrain the possible wavelengths or frequencies of vibration of the system.
`color{blue} ✍️`The system cannot oscillate with any arbitrary frequency (contrast this with a harmonic travelling wave), but is characterized by a set of natural frequencies or normal modes of oscillation. Let us determine these normal modes for a stretched string fixed at both ends.
`color{blue} ✍️`First, from Eq. (15.37), the positions of nodes (where the amplitude is zero) are given by `sin kx = 0` . which implies
`kx = nπ; n = 0, 1, 2, 3, ...`
Since `k = 2π//λ` , we get
`color{blue} ✍️`Clearly, the distance between any two successive nodes is `lamda/2` . In the same way, the positions of antinodes (where the amplitude is the largest) are given by the largest value of `sin kx `:
`|sin k x|= 1`
`color{blue} ✍️` which implies `color {purple} {kx = (n + ½) π ; n = 0, 1, 2, 3, ...}`
`color{blue} ✍️`With `k = 2π//λ`, we get
`color{blue} ✍️`Again the distance between any two consecutive antinodes is `lamda/2`. Eq. (15.38) can be applied to the case of a stretched string of length L fixed at both ends.
`color{blue} ✍️`Taking one end to be at `x = 0,` the boundary conditions are that `x = 0` and `x = L` are positions of nodes. The `x = 0` condition is already satisfied. The `x = L` node condition requires that the length L is related to λ by
`color{blue} ✍️`Thus, the possible wavelengths of stationary waves are constrained by the relation
`color{blue} ✍️`We have thus obtained the natural frequencies - the normal modes of oscillation of the system. The lowest possible natural frequency of a system is called its fundamental mode or the first harmonic.
`color{blue} ✍️`For the stretched string fixed at either end it is given by `nu = nu/(2L)`, corresponding to n = 1 of Eq. (15.42).
`color{blue} ✍️`Here `v` is the speed of wave determined by the properties of the medium. The `n = 2` frequency is called the second harmonic; `n = 3` is the third harmonic and so on. We can label the various harmonics by the symbol `νn \ \ ( n = 1, 2, ...).`
`color{blue} ✍️`Fig. 15.13 shows the first six harmonics of a stretched string fixed at either end. A string need not vibrate in one of these modes only.
`color{blue} ✍️`Generally, the vibration of a string will be a superposition of different modes; some modes may be more strongly excited and some less. Musical instruments like sitar or violin are based on this principle. Where the string is plucked or bowed, determines which modes are more prominent than others.
`color{blue} ✍️`Let us next consider normal modes of oscillation of an air column with one end closed and the other open. A glass tube partially filled with water illustrates this system.
`color{blue} ✍️`The end in contact with water is a node, while the open end is an antinode. At the node the pressure changes are the largest, while the displacement is minimum (zero).
`color{blue} ✍️`At the open end - the antinode, it is just the other way - least pressure change and maximum amplitude of displacement. Taking the end in contact with water to be `x = 0,` the node condition (Eq. 15.38) is already satisfied. If the other end `x = L` is an antinode, Eq. (15.39) gives
`color{blue} ✍️`The possible wavelengths are then restricted by the relation :
`color{blue} ✍️`The normal modes – the natural frequencies – of the system are
`color{blue} ✍️`The fundamental frequency corresponds to `n = 0`, and is given by `nu/(4L)` . The higher frequencies are odd harmonics, i.e., odd multiples of the fundamental frequency : `3 nu/(4L) , 5 nu/(4L)` , etc.
`color{blue} ✍️`Fig. 15.14 shows the first six odd harmonics of air column with one end closed and the other open. For a pipe open at both ends, each end is an antinode.
`color{blue} ✍️`It is then easily seen that an open air column at both ends generates all harmonics (See Fig. 15.15). The systems above, strings and air columns, can also undergo forced oscillations. If the external frequency is close to one of the natural frequencies, the system shows resonance.
`color{blue} ✍️`Normal modes of a circular membrane rigidly clamped to the circumference as in a tabla are determined by the boundary condition that no point on the circumference of the membrane vibrates.
`color{blue} ✍️`Estimation of the frequencies of normal modes of this system is more complex. This problem involves wave propagation in two dimensions.
`color{blue} ✍️` In a string, for example, a wave going to the right will get reflected at one end, which in turn will travel and get reflected from the other end. This will go on until there is a steady wave pattern set up on the string. Such wave patterns are called `"standing waves"` or stationary waves.
`color{blue} ✍️`To see this mathematically, consider a wave travelling along the positive direction of x-axis and a reflected wave of the same amplitude and wavelength in the negative direction of x-axis. From Eqs. (15.2) and (15.4), with φ = 0, we get:
`color{ blue } {y_1(x, t) = a sin (kx – ωt)}`
`color{blue} {y_2(x, t) = a sin (kx + ωt)}`
`color{blue} ✍️`The resultant wave on the string is, according to the principle of superposition:
`color {brown} { y (x, t) = y_1(x, t) + y_2(x, t)}`
`color {purple} { = a [sin (kx – ωt) + sin (kx + ωt)]}`
`color{blue} ✍️`Using the familiar trignometric identity
`color{purple} {Sin (A+B) + Sin (A–B) = 2 sin A cosB}` we get,
`color{blue} { y (x, t) = 2a sin kx cos ωt}`
........................ (15.37)`color{brown} {"Note"}` the important difference in the wave pattern described by Eq. (15.37) from that described by Eq. (15.2) or Eq. (15.4).
`color{blue} ✍️`The terms `kx` and `ωt` appear separately, not in the combination `kx - ωt.` The amplitude of this wave is `2a sin kx.`
`color{blue} ✍️`Thus in this wave pattern, the amplitude varies from point to point, but each element of the string oscillates with the same angular frequency `ω` or time period. There is no phase difference between oscillations of different elements of the wave.
`color{blue} ✍️`The string as a whole vibrates in phase with differing amplitudes at different points. The wave pattern is neither moving to the right nor to the left. Hence they are called `"standing or stationary waves."`
`color{blue} ✍️`The amplitude is fixed at a given location but, as remarked earlier, it is different at different locations. The points at which the amplitude is zero (i.e., where there is no motion at all) are nodes; the points at which the amplitude is the largest are called `"antinodes."`
`color{blue} ✍️`Fig. 15.12 shows a stationary wave pattern resulting from superposition of two travelling waves in opposite directions.
`color{blue} ✍️`The most significant feature of stationary waves is that the boundary conditions constrain the possible wavelengths or frequencies of vibration of the system.
`color{blue} ✍️`The system cannot oscillate with any arbitrary frequency (contrast this with a harmonic travelling wave), but is characterized by a set of natural frequencies or normal modes of oscillation. Let us determine these normal modes for a stretched string fixed at both ends.
`color{blue} ✍️`First, from Eq. (15.37), the positions of nodes (where the amplitude is zero) are given by `sin kx = 0` . which implies
`kx = nπ; n = 0, 1, 2, 3, ...`
Since `k = 2π//λ` , we get
`color{blue} {x = (nλ)/2 ; n = 0, 1, 2, 3, ...}`
....................... (15.38)`color{blue} ✍️`Clearly, the distance between any two successive nodes is `lamda/2` . In the same way, the positions of antinodes (where the amplitude is the largest) are given by the largest value of `sin kx `:
`|sin k x|= 1`
`color{blue} ✍️` which implies `color {purple} {kx = (n + ½) π ; n = 0, 1, 2, 3, ...}`
`color{blue} ✍️`With `k = 2π//λ`, we get
`color { blue} { x = (n + ½) lamda/2 ; n = 0, 1, 2, 3, ...}`
................................ (15.39)`color{blue} ✍️`Again the distance between any two consecutive antinodes is `lamda/2`. Eq. (15.38) can be applied to the case of a stretched string of length L fixed at both ends.
`color{blue} ✍️`Taking one end to be at `x = 0,` the boundary conditions are that `x = 0` and `x = L` are positions of nodes. The `x = 0` condition is already satisfied. The `x = L` node condition requires that the length L is related to λ by
`color{ blue} { L = (n lamda)/2 ; n = 1, 2, 3, ...}`
................... (15.40)`color{blue} ✍️`Thus, the possible wavelengths of stationary waves are constrained by the relation
`color{blue} {lamda = (2L)/n ; n = 1, 2, 3, …}`
..................... (15.41)`color{blue} {nu = (n nu )/(2L) ," for "n = 1, 2, 3}`,
.......................(15.42)`color{blue} ✍️`We have thus obtained the natural frequencies - the normal modes of oscillation of the system. The lowest possible natural frequency of a system is called its fundamental mode or the first harmonic.
`color{blue} ✍️`For the stretched string fixed at either end it is given by `nu = nu/(2L)`, corresponding to n = 1 of Eq. (15.42).
`color{blue} ✍️`Here `v` is the speed of wave determined by the properties of the medium. The `n = 2` frequency is called the second harmonic; `n = 3` is the third harmonic and so on. We can label the various harmonics by the symbol `νn \ \ ( n = 1, 2, ...).`
`color{blue} ✍️`Fig. 15.13 shows the first six harmonics of a stretched string fixed at either end. A string need not vibrate in one of these modes only.
`color{blue} ✍️`Generally, the vibration of a string will be a superposition of different modes; some modes may be more strongly excited and some less. Musical instruments like sitar or violin are based on this principle. Where the string is plucked or bowed, determines which modes are more prominent than others.
`color{blue} ✍️`Let us next consider normal modes of oscillation of an air column with one end closed and the other open. A glass tube partially filled with water illustrates this system.
`color{blue} ✍️`The end in contact with water is a node, while the open end is an antinode. At the node the pressure changes are the largest, while the displacement is minimum (zero).
`color{blue} ✍️`At the open end - the antinode, it is just the other way - least pressure change and maximum amplitude of displacement. Taking the end in contact with water to be `x = 0,` the node condition (Eq. 15.38) is already satisfied. If the other end `x = L` is an antinode, Eq. (15.39) gives
`color{blue} { L = ( n + 1/2 ) lamda/2 }" for "n = 0, 1, 2, 3, …`
`color{blue} ✍️`The possible wavelengths are then restricted by the relation :
`color{blue} {lamda = ( 2L)/ (( n + 1/2 )) ," for "n = 0, 1, 2, 3,... }`
.......................(15.43)`color{blue} ✍️`The normal modes – the natural frequencies – of the system are
`color{blue} {nu = ( n + 1/2 ) nu /(2L) ; n = 0, 1, 2, 3, }`
................ (15.44)`color{blue} ✍️`The fundamental frequency corresponds to `n = 0`, and is given by `nu/(4L)` . The higher frequencies are odd harmonics, i.e., odd multiples of the fundamental frequency : `3 nu/(4L) , 5 nu/(4L)` , etc.
`color{blue} ✍️`Fig. 15.14 shows the first six odd harmonics of air column with one end closed and the other open. For a pipe open at both ends, each end is an antinode.
`color{blue} ✍️`It is then easily seen that an open air column at both ends generates all harmonics (See Fig. 15.15). The systems above, strings and air columns, can also undergo forced oscillations. If the external frequency is close to one of the natural frequencies, the system shows resonance.
`color{blue} ✍️`Normal modes of a circular membrane rigidly clamped to the circumference as in a tabla are determined by the boundary condition that no point on the circumference of the membrane vibrates.
`color{blue} ✍️`Estimation of the frequencies of normal modes of this system is more complex. This problem involves wave propagation in two dimensions.
