Standing Waves in Strings and Organ Pipes

Many musical instruments consist of an air column enclosed inside a hollow metal tube. Sometimes the metal tube may be more than a metre in length; it is often curved upon itself one or more times to conserve space. 

 Organ Pipe 

The mouthpiece was not designed; pipes were set in broad daylight spaces like the

church. The organ lines would resound with similar frequencies and produce a noisy

clear sound. These should be visible in numerous old places of worship worldwide.

The organ pipe is the instrument where sound is delivered by setting and air segment

into vibrations.

We will examine the two kinds of Organ pipes that are

(1) Closed organ pipes-shut down toward one side while open at the opposite end.

(2) Open organ pipe – open at the two finishes.

This module will talk about the development of standing waves in shut and open organ pipes.

Waves in a Closed Organ Pipe

Analytical treatment of Standing waves in a closed organ pipe

1.  Let us consider normal oscillation modes of an air column with one end closed and the other open. 
2. A glass tube partially filled with water can illustrate this system.

1. Closed organ pipe-one finish of the line is open, and the opposite end is shut.

1. Assuming that air is blown softly at the open finish of the shut organ pipe,
2. the air section sets into vibrations and wave accordingly produced gets reflected from the shut end,
3. the course of movement of particles changes, the relocation is zero at the shut end.

1. The uprooting is greatest at the open end, implying a hub at the shut end and an antinode at the open end.

Consider a cylindrical pipe of length L placed vertically with its closed end (node) at x = 0

and antinode at x = L

Now consider a sound wave travelling along the tube and a reflected wave of the same amplitude

and wavelength in the opposite direction.

The wave travelling down can be represented as

 𝒚𝟏

(𝒙, 𝒕) = 𝒂 𝐬𝐢𝐧(𝝎𝒕 − 𝒌𝒙)

The wave travelling along negative direction can be represented as

𝒚𝟐

(𝒙, 𝒕) = −𝒂 𝐬𝐢𝐧(𝝎𝒕 + 𝒌𝒙)

The resultant wave on the string is, according to the principle of superposition:

𝑦(𝑥,𝑡) = 𝑦1

(𝑥,𝑡) + 𝑦2(𝑥,𝑡)

 = 𝒂 [𝐬𝐢𝐧(𝝎𝒕 − 𝒌𝒙) − 𝐬𝐢𝐧(𝝎𝒕 + 𝒌𝒙)]

Using the familiar trigonometric identity

sin (A+B) – sin (A–B) = 2 sin B cos A, we get,

y (x, t) = -(2a sin kx) cos ωt

This equation represents a standing wave.

The amplitude of this wave is 2a sin kx.

Nodes:

 Where the amplitude is zero (i.e., where there is no movement of particles by any means) are nodes.

 Antinodes: 

Where the amplitude of the molecule is the highest are called antinodes.

Standing Waves in Pipes

Pipes are used to make a variety of musical instruments. Wind instruments are the name for these types of instruments. A flute, for example, is a single pipe with many holes that may be opened to modify the effective length of the pipe. The effective length is adjusted by sliding a tube in or out. On the other hand, in a pipe organ, several different pipes of varying lengths are employed, each with its fundamental frequency. The sound of the pipe is determined by the effective length of the tube, which is common to all of these instruments. Unlike a string instrument, where a vibrating string creates a sound wave in the air, a wind instrument’s wave is already a sound wave in a column of air, part of which escapes to produce audible sound. Standing waves formed by reflected waves, like strings, define the fundamental frequency of the sound wave produced by a particular pipe. It’s important to note that pipes might have both ends open or one end open and the other closed. The open end acts as a free end for a sound wave, while the closed end acts as a fixed end. Because of the distinct standing waves created by the different reflections, a pipe with only one end open sounds significantly different than a pipe with both ends open, even though the tubes are the same length.

Standing waves formed by a pipe with both ends open always have an antinode at each end. The following frequencies cause standing waves in such a pipe:

fn=vn=nv2L

where n is an integer and L is the pipe’s effective length.

Standing waves for a pipe with just one open end include anti-nodes at the open end and nodes at the closed end because an open-end behaves like a free end, and closed-end acts as a fixed end. Standing waves with an odd integer number of quarter-wavelengths fit in the pipe can meet this requirement. As a result, a new equation for standing-wave frequencies emerges.

The first mode of vibration:

Two antinodes develop at two open ends of the organ pipe, and one node forms in the middle. If the pipe length is ‘l,’ and the wavelength of the wave emitted in this mode of vibration is

 l1=24, =2l.

If ‘μ’ is the velocity of sound and the wave’s frequency in this mode of vibration. Then f=V=V2l

The second mode of vibration:

In the open organ pipe’s second phase of vibration, antinodes occur at both ends and two nodes in the middle.

If ‘l’ is the pipe length and 1 is the wavelength of the wave emitted in this mode of vibration, then

l=`1

1=l

If μ be the velocity of sound and to be the frequency of the wave in this mode of vibration, then 

f1=Vl

f1=2f

The third mode of vibration:

Four antinodes and three nodes are created in the third phase of vibration in the open organ pipe. If l is the pipe length and 2 is the wavelength of the wave emitted in this mode of vibration,

l=322

2=2l3

Let f be the sound velocity on this vibration mode, and f2 be the wavelength frequency. Then

f2=2v32

f2=3f

The frequency of a wave for the 4th, 5th…………….. modes of vibration are f3 = 4f0; the frequency of 4th, 5th harmonics is f4 = 5f0……………. As a result, both odd and even harmonics are present in open organ pipes. The hollow organ’s tone is then quite thunderous.

Conclusion

These pipes are also known as organ pipes and are divided into two types: open mouth and closed mouth organ pipes. The incident sound waves are reflected and superimposed in these pipes, resulting in stationary waves. Let L denote the length of the organ pipe, and v denote the sound velocity. Standing waves can be longitudinal or transverse. The disruption is localised to a specific zone between the wave’s beginning and reflecting points. Beyond this specific zone, there is no forward mobility of the disturbance from one particle to the next, and so on. The outcome of a sound wave bouncing between two or more surfaces is a standing wave, which accentuates one specific frequency that you hear when the waves reinforce each other—the wave’s phase shifts when it bounces off the surface.