Physics - Vibrations & Waves

Ok I have written the question down below, forgive me for not knowing how to use LaTeX. I am struggling on question ii) c & d

Also could someone please confirm that the frequency received in part ii)b) is the same as the derived in part ii)a

This is not homework or anything of the assessed kind.

It is strictly for the purpose of revision.

Thank you

A scientist designs a simple underwater sonar system. A metal tube of length L,
with both ends open, is lowered into the sea from a boat. A mechanical oscillator
is used to create sound waves in the tube. By pulsing the oscillator, the tube can be
made to emit sound wave pulses of frequency f = 2 kHz. A microphone picks up
any reflected echoes from objects in the sea.

(a) Sketch the 1st, 2nd, 3rd and 4th modes (ie. harmonics) of a one-dimensional
standing wave between x = 0 and x = L where the boundary conditions are
such that there is an antinode at x = 0 and an antinode at x = L. [2 marks]

(b) In the tube, longitudinal sound waves travelling in the positive x-direction
ψ+(x, t) and in the negative x-direction ψ−(x, t) have the general form:
ψ+(x, t) = A cos(ωt − kx + φ)
ψ−(x, t) = A cos(ωt + kx + φ)
Let the waves in the tube be reflected at x = 0 and x = L according to:
ψ+(x = 0,t) = ψ−(x = 0,t)
ψ+(x = L,t) = ψ−(x = L,t)

By combining ψ+(x, t) and ψ−(x, t), show that standing sound waves in the
tube have allowed wavelengths given by:
λn = 2L

n
where n = 1, 2, 3, 4, …. [4 marks]

(c) The phase velocity of sound waves in water is given by:
v = sqrt(B/ρ)
where B is the bulk modulus and ρ is the density. If B = 2.18 × 109 Pa and
ρ = 103 kgm−3, what length of tube L would be required to give a 1st harmonic
equal to f ?[2 marks]

ii) (a) A shark swims towards the boat at a velocity of uShark = 1ms−1. The boat is
stationary in the water. A pulse from the sonar of wavelength λ and frequency
f reaches the shark. By considering the compression of the wavelength as the
shark swims into it, show that the frequency f received by the shark is given
by:
f ' = ((v + uShark)f)/v
where v is the phase velocity of the sound waves given above by the equation
in part (i)(c). [4 marks]

(b) The pulse bounces back from the shark to the boat. What frequency f is
detected by the scientist due to the motion of the shark? [2 marks]

(c) At a depth of 10m below surface the water suddenly becomes less salty so that
its density changes to 0.9×103 kgm−3. Beyond what angle to the vertical would
the sonar no longer detect anything in this less salty water? The shark is at a
depth < 10 m. The scientist receive two echoes from the shark. What is the
minimum horizontal distance the shark has to be from the boat? [2 marks]

(d) The sound wave pulse from the sonar has a total average power of 1W. Assume
that this is emitted in all directions. Let the shark be at a horizontal distance of
20m and a depth of 5m. Assume that it has an effective surface area of 1m2
perpendicular to the direction of travel of the sound wave. What intensity will
the microphone detect in the returning echo from the shark? [2 marks]
[TOTAL 18 marks]

Physics - Vibrations & Waves

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