Waves in Pipes and Strings

(a) Organ pipe: An organ pipe is a cylindrical tube of wood or metal in which sound is produced by setting the air column inside it into vibration (by blowing air at one end). A pipe closed at one end is a closed organ pipe; a pipe open at both ends is an open organ pipe.

(b) Modes of vibration in a closed organ pipe

When air is blown at the open end, the wave reflects from the closed end and stationary waves form. A node always forms at the closed end and an antinode at the open end. Let $L$ be the length of the pipe and $v$ the speed of sound.

 Fundamental        First overtone       Second overtone
 |N        A        |N    A   N   A      |N A  N  A  N  A
 |=========         |=============       |================
 L = λ/4            L = 3λ/4             L = 5λ/4

1. Fundamental (first harmonic): one node (closed end) and one antinode (open end).

2. First overtone (third harmonic): two nodes and two antinodes.

3. Second overtone (fifth harmonic): three nodes and three antinodes.

In general $f_n = (2n-1)\dfrac{v}{4L}$, so

A closed pipe produces only odd harmonics.

(c) End correction: The antinode at an open end is not formed exactly at the end but slightly outside it, because the air just outside the mouth also vibrates. The extra distance is the end correction, $e \approx 0.6r = 0.3d$ ($r$ = radius, $d$ = diameter of the pipe). Effective length: $L + e$ for a closed pipe and $L + 2e$ for an open pipe.

(a) Stationary waves: When two identical progressive waves (same amplitude, frequency and speed) travel in opposite directions along the same line, they superpose to give a wave pattern that does not advance. It has fixed points of zero displacement (nodes) and maximum displacement (antinodes). This is a stationary wave.

(b) Formation in an open organ pipe: When air is blown into one end, a compression travels down the pipe and is reflected at the other open end (as a rarefaction). The incident and reflected waves, having the same frequency and amplitude, travel in opposite directions and superpose to form stationary waves. Both ends are free, so antinodes form at both open ends.

(c) Wave patterns and frequencies (length $L$, speed of sound $v$)

 Fundamental (1st harmonic)       First overtone (2nd harmonic)
 A          N          A          A     N     A     N     A
 )==========X==========(          )=====X=====()=====X=====(
 |<-------- L = λ/2 ------>|      |<--------- L = λ ------->|

Fundamental: antinodes at both ends and one node in the middle.

First overtone: three antinodes and two nodes.

(In general $f_n = \dfrac{nv}{2L}$; an open pipe gives all harmonics.)

An organ pipe is a cylindrical tube of wood or metal in which sound is produced by making the air column inside it vibrate, usually by blowing air at one end. If one end is closed it is a closed organ pipe; if both ends are open it is an open organ pipe.

Ratio $L_A : L_B = 8 : 7$

Justification:

Open pipe A: the $n$th overtone is the $(n+1)$th harmonic, so the third overtone is the 4th harmonic:

Closed pipe B: only odd harmonics occur ($1, 3, 5, 7, \dots$), so the third overtone is the 7th harmonic:

In resonance, $f_A = f_B$:

i) Nodes and antinodes

   A         N         A         N         A
   )=========X=========()========X=========(
   |<------------------ L ----------------->|
   0        L/4       L/2      3L/4         L

Antinodes (A) at both open ends and at the centre; nodes (N) at $L/4$ and $3L/4$.

ii) Frequency of this mode

This is the first overtone (second harmonic) of an open pipe.

1. Distance between two consecutive antinodes $= \lambda/2$. There are two such segments in the pipe:

2. Using $v = f\lambda$:

This is twice the fundamental frequency $\dfrac{v}{2L}$.

The speed of a transverse wave on a string is

where $\mu$ is the mass per unit length. For a string of radius $r$ and density $\rho$:

So, for the same tension and material, $v \propto \dfrac{1}{r}$.

If the radius is doubled, $v' = \dfrac{v}{2}$. The wave velocity becomes half.

Given: $L = 1.5$ m, $\rho = 7800$ kg m$^{-3}$, $Y = 2\times10^{11}$ N m$^{-2}$, strain $= 1\% = 0.01$

i) Stress

ii) Frequency of the second mode

Tension $T = \text{stress} \times A$ and mass per unit length $\mu = \rho A$, so the wave speed is

In the second mode (second harmonic) the string vibrates in two loops, so $L = \lambda$:

Stress = $2\times10^9$ N m$^{-2}$; frequency of second mode $\approx$ 338 Hz.

i) The figure shows antinodes at both ends and at the middle, with two nodes in between (3 antinodes, 2 nodes). This is the first overtone (second harmonic) of the open pipe.

ii) One full wavelength fits in the pipe, so $L = \lambda$. Hence

where $v$ is the speed of sound and $L$ the length of the pipe.

(b) $\dfrac{\lambda}{4}$

Consecutive nodes are $\lambda/2$ apart and an antinode lies midway between them, so the node-to-antinode distance is $\lambda/4$.

a) Why end correction is necessary

In theory, the antinode is taken exactly at the open end of the pipe. In practice, the air just outside the open end also takes part in the vibration, so the antinode forms a little outside the end. The vibrating air column is therefore longer than the pipe. If we use only the geometrical length $L$, the calculated frequencies come out too high. So we add the end correction $e \approx 0.6r$ (r = radius):

b) Knowing when the vessel is about to fill

The air above the water in the vessel acts as a closed organ pipe (closed by the water surface). The falling water makes this air column vibrate, and the sound we hear is mostly its fundamental frequency

As water fills the vessel, the length $l$ of the air column decreases, so $f$ increases. The pitch of the sound becomes higher and higher; a very shrill sound tells us the vessel is almost full.

c) Type of pipe

Only odd harmonics are present, so the pipe is closed (fundamental 200 Hz).

d) Modes of vibration in a closed organ pipe

A node forms at the closed end and an antinode at the open end. Let length = $L$, speed of sound = $v$.

 Fundamental        First overtone       Second overtone
 |N        A        |N    A   N   A      |N A  N  A  N  A
 |=========         |=============       |================
 L = λ/4            L = 3λ/4             L = 5λ/4

1. Fundamental: $L = \dfrac{\lambda_1}{4} \Rightarrow f_1 = \dfrac{v}{4L}$
2. First overtone: $L = \dfrac{3\lambda_2}{4} \Rightarrow f_2 = \dfrac{3v}{4L} = 3f_1$
3. Second overtone: $L = \dfrac{5\lambda_3}{4} \Rightarrow f_3 = \dfrac{5v}{4L} = 5f_1$

So $f_1 : f_2 : f_3 = 1 : 3 : 5$; a closed pipe gives only odd harmonics.

(i) Harmonics and overtone

Harmonics are the frequencies that are integral multiples of the fundamental frequency ($f, 2f, 3f,\dots$). Overtones are the frequencies of vibration higher than the fundamental; the first overtone is the next higher mode after the fundamental.

(ii) Frequency of the mode

The figure shows two loops, so the length of the string is one full wavelength: $l = \lambda$, i.e. $\lambda = l$. With the wave speed $v = \sqrt{T/\mu}$:

(this is the second harmonic, $f = 2f_0$ where $f_0 = \frac{1}{2l}\sqrt{T/\mu}$).

(iii) Tension increased four times

Since $f \propto \sqrt{T}$,

The frequency of this mode becomes double.

(d) 990 Hz

A pipe closed at one end supports only odd harmonics: $f_n = (2n-1)f_1$. The first mode is $f_1 = 330$ Hz, so the second mode (the first overtone) is the third harmonic:

$f = 3f_1 = 3\times 330 = 990\ \text{Hz}$

(a) For the open pipe of length $L_o$, the first harmonic is the fundamental:

$f = \frac{v}{2L_o}$

For the closed pipe of length $L_c$, the first overtone is the third harmonic:

$f = \frac{3v}{4L_c}$

Setting them equal:

$\frac{v}{2L_o} = \frac{3v}{4L_c} \;\Rightarrow\; L_c = \frac{3}{2}L_o$

So the closed pipe is 1.5 times as long as the open pipe.

(b) End correction is the small extra length (about $0.6r$ for a pipe of radius $r$) that must be added to the actual length of a pipe, because the antinode forms slightly outside the open end.

(c) A string of length $L$ fixed at both ends vibrates with nodes at both ends and in standing-wave patterns of

$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}, \qquad n = 1, 2, 3,\dots$

The mode $n=1$ is the fundamental, with one loop and frequency $f_1$. The mode $n=2$ is the second harmonic (first overtone) with two loops and frequency $2f_1$. In general the $n$th harmonic has $n$ loops and $n-1$ intermediate nodes, and all integer harmonics are present.

(a) Frequency of the tuning fork

Given: $l_1 = 75$ cm, $l_2 = 75 - 0.5 = 74.5$ cm, beats = 3 per second.

For a sonometer wire with fixed tension, $f \propto \dfrac{1}{l}$, so $fl =$ constant.

1. At 75 cm the wire is in resonance, so its frequency equals the fork frequency $f$.
2. A shorter wire has a higher frequency, so at 74.5 cm the wire frequency is $f + 3$.
3. Using $f_1 l_1 = f_2 l_2$:

Frequency of the tuning fork = 447 Hz.

(b) Speed of a progressive wave on the string

Given: $f = 120$ Hz, length between vibrator and pulley $= 75$ cm $= 0.75$ m, 6 loops.

Each loop is $\lambda/2$, so

Speed of the wave = 30 m/s.

(i) Harmonics: Harmonics are the tones whose frequencies are whole-number multiples of the fundamental (lowest) frequency, i.e. $f, 2f, 3f, \dots$. The fundamental itself is the first harmonic.

(ii) Frequency of the flute

Given: $L = 30$ cm $= 0.30$ m, open at both ends, first harmonic, $v = 330$ m s$^{-1}$.

For an open pipe in the fundamental mode, $L = \lambda/2$:

Frequency = 550 Hz.