# Useful Formulae

40 SOUND harmonic wavelengths and passing sirens For a string fixed at two points distance L apart, as shown, the harmonic wavelengths which fit the string are given by n for n 1,2,3 ... Only 1 and the first few overtonesmay actually be audible. If WTis the wave velocity in the string,Tits tension in Newtons and the mass per unit length in kgm,then the basic frequencies, n,in cycles per second are n The tone of an ambulance siren appears to change as it passes. This is known as the Doppler Effect. If an observer is moving at velocity votowardsa source of frequencyfs,which is itself moving at velocity vs towards the observer,then for a wave of speed cthe observed frequency fois fo fs Note if objects recede rather than approach,use negative values for vo and vs the speed of sound is 331.45 msec. If two very similar tones,frequencies f1 and f2,are played together, a beat frequency fbeat f2 f1 may be heard. c vs 2L n WT T WT n 2L n 2L n c vo 41