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Characteristics of Sound Waves

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Transverse and Longitudinal Waves

Classification of waves is according to the direction of propagation. In transverse waves the particles vibrate perpendicular to the direction of propagation (for example, a vibrating string or a water wave) In longitudinal waves the particles vibrate in the direction of propagation (for example, an oscillating spring or sound waves)

The subject of sound is known in physics as acoustics. One differentiates sound according to the frequency as infrasonic, audible sound, ultrasonic and hypersonic. Sonic infrasound is a sound too low for humans to hear, below 20 Hertz. Ultrasound is a sound too high for humans to hear, above 20,000 Hertz.

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We can obtain an equation of motion (a "wave equation") for a particle on a stretched string by applying F=ma to a little piece of string. When we do this, we find that a solution is any function whose argument is x + vt or x - vt. The exact nature of the function f(x - vt) depends on how the string is wiggled.When the end of a string is wiggled like a harmonic oscillator, the transverse displacement of the string is given by

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air pressure

time

longitudinal wave

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Speed of SoundA sound wave can be represented by placing a long coiled spring on a horizontal table. If one moves the end back and forth harmonically, regions of compression and rarefaction travel along the spring. The speed of sound accounts for these changes in pressure and can be written as an equation depending on its elastic characteristics.

For which B is the bulk modulus and is the mass density of the medium in which the sound is propagating.

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Problems

1.) For steel the bulk modulus is 60 GPa and its density is 8 kg·m−3.  What is the speed of sound in steel?

Solution

2.) A wave is described as where k = 2.14 rad/m and ! = 3.6 rad/s. Determine the amplitude, wavelength, frequency, and speed of the wave. SolutionThe amplitude is given. It is 2.10 cm. The wavelength ¸ is

The frequency f is

The speed is

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Superposition of Waves

If wave displacements are added together the resulting wave can show either constructive or destructive interference. If two waves of the same velocity and wave-lengths are travelling in the same direction, they will interfere. If they are in phase they interfere constructively and result in a stronger wave. If they are out of phase andhave the same amplitude, they cancel each other out(destructive interference ).

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Superposition of Waves

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Two waves of the same phase Two waves with equal but opposite phase

Constructive Interference Destructive Interference

Standing WavesA standing wave is the result of two waves of the same frequency and amplitude moving in opposite directions to each other. A mechanical example is a string that one wiggles up and down and produces a propagating wave. If one end is fixed, the wave will be reflected. As a result, one can’t observe a propagating wave and instead the string experiences an oscillation in one place. The antinodes will remain fixed and the nodes will oscillate with a larger amplitude. The distance between two nodes or antinodes is the half wavelength of the original wave. Thus for nodes we have the equation

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NodesThe distance L of a crest from the center is a half multiple of the wavelength. with n = 1, 2, 3 ...

AntinodesThe distance L of a node from the center is a multiple of the wave length plus a quarter

with n= 1, 2, 3 ...

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Harmonics of Standing Waves

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1st Harmonic

2nd Harmonic

3rd Harmonic

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BeatsIf two travellng waves have slightly different frequencies, they interfere and produce a phenomen called beats.

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The Doppler Effect

The doppler effect is the change in perceived or measured frequency as the observer and the source travel relative to one another of any kind of wave.

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If the observer and the source move towards each other, the frequency perceived by the observer increases, if they move apart the frequency decreases. For example, the sound of an ambulance gets higher as it gets nearer and the tone gets lower as it moves away. Observer at rest and source of the signal moves

Observer moves and source of the signal at rest. The spacing of the crests of the waves coming towards you is