Learning Object for Phys 101

1. Whalesare capable of makingsoundsunderwater,andsome evenconsiderthem tosingsongs.Imagine
somewhere inthe ocean,amotherandfatherwhale are bothharmonizingand singingtheirfavourite
lullaby tofindtheir offspring.Theirvoicesare oscillatinginphase andhave a frequencyof 200 Hz in
whichthe speedof soundunderwaterisapproximately340 m/s. The calf is7 m fromthe motherwhale
and 9 m fromthe fatherwhale.
a) What isthe phase difference inthe soundwavesfrom the twoparentsatthe calf’scurrent
location?
b) As the parentscontinue tosing,the calf starts to singback butat a frequencyof 210 Hz.What is
the beat frequency heard?
Solution:
a) As explainedin PhysicsforScientistsandEngineers - 1e byHawkes,Iqbal,Mansour,Milner-
Bolotin,andWilliams,the phase difference isthe difference betweenthe phasesattwopoints.
The formulato calculate the phase difference is:
∆ 𝝓 = 2 π ∆x = 2 π f ∆x = (2 π) * (200 Hz) * (9 m – 7 m) = 7.39 radians.
λ v 340 m/s
b) In the pre-reading,youlearnedthatabeatis a variationinamplitude. The rate atwhichthe
amplitude variesisproportional tothe frequencydifference.Inthiscase,we are modulatingthe
amplitude,where itwill be large forboththe positiveandnegative crests. The amplitude
modulations will have the same frequencyasthe differencebetweenthe twofrequencies.
210 Hz – 200 Hz = 10 Hz.
Althoughitisharderto detect,itis still possible forbeatsof a10 Hz modulationtobe heard.As
we know,the lowerthe frequencydifference,the easieritistodetectand hearthe beats.