Standing+Waves

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**The principle of superposition** - The net displacement of the medium / particles (through which waves travel) due to the superposition is equal to the sum of individual displacements (produced by each wave).


 * Superposition of Waves** The principle of superposition may be applied to waves whenever two (or more) waves travelling through the same medium at the same time. The waves pass through each other without being disturbed. The net displacement of the medium at any point in space or time, is simply the sum of the individual wave dispacements. This is true of waves which are finite in length (wave pulses) or which are continuous sine waves.

Animation of superposition and standing waves

**Standing waves** - The periodic disturbance in a medium resulting from the combination of two waves of equal frequency and intensity travelling in opposite directions. The wave profile doesn’t progress. All points in between two nodes on a standing wave are in phase. (whereas points on a progressive wave that are closer than one wavelength are all out of phase.)

Node: A place on the wave that doesn’t move at all where the displacement is zero. Antinode: A place on the wave where the amplitude is the maximum.

Standing waves applet

Simulation: Principle of superposition

Principle of superposition 2

Stationary waves

Reflection on a sine wave

**Stringed instruments**
Harmonics on a string The speed of a wave in a string. //T=// the tension, μ= The mass per unit length

Harmonic motion, as of a violin string, can be analyzed as the sum of harmonic frequencies or harmonics, each of which is itself a kind of harmonic motion. All the possible standing waves are produced.

**Standing waves in closed pipes.**
When a standing wave is formed in a closed pipe, only **odd harmonics** are formed.



**Standing waves in open pipes.**
When a standing wave is formed in a open pipe, **all the harmonics** are formed.

Wikipedia - [|harmonics on guitar string] Walter Fendt - [|Standing waves applet] Surendranath - [|Harmonics] Harmonic applet on[| string] Harmonics on a [|violin string] [|Physics of Guitar strings] SMU - [|Standing waves in a closed & open pipe]

**Examples** ( Consider the speed of sound in air as 340m/s )
Answer 1. Pipes: The fundamental frequency of a pipe closed at one end is //f//. A pipe of the same length but open at both ends has a fundamental frequency (first harmonic) of A.(1/2)//f// B. //f// C. 2//f// D.4//f//

Answer 2. Organ pipes: What will be the fundamental frequency and first three overtones for a 26 cm long organ pipe if it is (a) open (b) closed

Answer 3. Flute: A flute is designed to play middle C (262Hz) as the fundamental frequency when all the holes are covered. Approximately how long should the distance be from the mouthpiece to the far end of the flute?

Answer 4. Wind noise frequencies: Wind can be noisy - it can "moan" in chimneys ( length:3m ). What is causing the noise, and about what frequency would you expect to hear?

Answer 5. Calculate the first harmonic produced when a standing wave is formed in a closed pipe of length 50cm.

Answer 6. The air in a closed pipe is made to vibrate by holding a tuning fork of frequency 256Hz over its open end. As the length of the pipe is increased, loud notes are heard as the standing wave in the pipe resonates with the tuning fork. (a) What is the shortest length that will cause a loud note? (b) If the pipe is 1.5 m long, how many loud notes will you hear as the plunger is withdrawn?