Saturday, 29 October 2016

acoustics - Why is this equation true for the sound pressure a loudspeaker creates?


In an answer, there is this equation:


p=ρSD2πra


This describes that the sound pressure (p) is proportional to the acceleration (a) of the cone of a loudspeaker.



(ρ is the density of air, SD is the surface area of the cone, and r is the distance from the cone)


I wonder, where does this equation come from? What's the theory behind it?



Answer



To derive this result, there are various possible starting points. Probably the most rigorous approach is to start from the pressure due to a baffled rigid piston. In this case, at a distance r along the axis of the piston we have


p(r,t)=ρc(1eikϕ)v(r,t),


where ρ is the density of air, c is the speed of sound,


ϕr2+r20randv(r,t)=v0ei(ωtkr).


Above, v is the (oscillating) piston velocity and r0 is the piston radius. This textbook result can be found, for example, in Section 7.4 of Kinsler (4th Ed). Note that the motion is assumed to be harmonic, such that ω is the oscillation frequency and k is the wavenumber such that ω=ck. In the far-field, for which ϕ1, this reduces to


p(t)=iρckϕv(t).


The presence of the i shows that the velocity is out of phase with the pressure. Since the motion is harmonic, we can rewrite this in terms of the acceleration using a=tv=iωv. Then, the complex pressure becomes



p(t)=ρϕa(t).


Taking rr0 gives ϕr20/(2r), or


p(t)=ρr202ra(t).


The area of the cone is SD=πr20, so we are left with


p(t)=ρSD2πra(t).


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