Power in a Sound Wave

Sound originates when a vibrating source disturbs the air in a quasi-periodic fashion and sends a sound wave travelling through it. Another way of saying this is that a sound source radiates acoustical energy which is transferred to the medium (air) and this energy propagates through the air in the form of a sound wave.

Our ears are not very sensitive at all to the total acoustical energy which reaches them. They are however sensitive to the rate at which the energy arrives. This rate is what determines loudness. The rate at which an instrument radiates acoustical energy is the instrument's acoustic power output. (Power is the rate of doing work.) The unit of power is the watt[1].

A musical instrument is a transformer. It is supplied with power by a performer or electricity. Much of this power is used to overcome friction/resistance and is "wasted" in other ways. Only a small proportion is transferred into musical sound. A pianist may use energy at the rate of 200 watts in a very loud passage without more than 0.4 watt being radiated as sound.

Table 2.1. Power radiated as sound by various musical instruments
(maximum loudness unless otherwise stated).

It can be observed from Table 2.1 above that the ratio of the of loudest to softest sounds is approx. 20,000,000 : 1. It can thus be appreciated that our ears are very sensitive to the rate at which energy arrives at the eardrum, i.e. they are sensitive to acoustical power. It is the amplitude of the eardrum oscillations which leads to the sensation of loudness and this amplitude is directly related to the average pressure variation *p, of the incoming sound wave.

[1] 1 watt is the power required to raise 0.45 Kg 215 mm/sec (approx.) A man doing hard continuous labour develops about 100 watts - the power required to keep a 100W electric light bulb burning.

[2 ] It would take thus approximately 2 million people in conversation to keep a 50 Watt electric light blub burning.

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