In the sixth series in our Brainwaver posts , we are intruding into some ‘ novel ‘ territory. Anyone who has read literature scenarios dealing with vast landscapes and mountain scenes will remember the author musings regarding the unsual nature of sounds and its effects being mentioned with thrilling accroutrements. And which fan of Sherlock Holmes will forget the chilling sound effects reverberating in the environs of the ‘Hound of the Baskervilles’ ?
The Case of the Distant Murmurs -
Well , coming to the point , it is not unusual to hear even faint sounds / murmurs originating from a distant place , in plain landscapes and hilly countryside especially when a wind is blowing towards you from the area of the sound source.
( a ) What is the reason behind this…. is it because the velocity of sound is enhanced by the blowing wind ? (b) Wind blowing across the ground has a vertical velocity gradient given by the formula v = k y² , where y is the height above the ground and k is a constant which depends on the wind speed outside the boundary layer where the parabolic velocity profile is a good approximation. Now , for a given value of k and speed of sound vs , find the distance s , downwind from a sound source , where the maximum enhancement of sound intensity happens ? Assume that the sound rays follow low , arc like paths represented by y = h sin ( π x / s ) . (c) Even without any wind , one is able to hear more clearly , sounds coming across a water body like a lake – what is happening here ?
Ans . (a) The clarity of the sounds / murmurs heard cannot be simply attributed to the wind blowing in the observer’s direction , as , if this were the case , then any observer in any path of the wind’s direction would hear the sounds with the same clarity. Hence , one has to look at the refractive effects of sound brought about by the variation of the sound velocity, with respect to a stationary observer , at different points of the medium. The key factors to consider here are the velocity gradient and the temperature gradient of the moving wind. Now , the velocity of compressible waves in a gaseous medium varies with temperature T as √ T . Variations also occurs if the velocity of the medium itself varies. Now , the refraction of sound waves will change the direction of its wavefront . Considering the surface of the earth in contact with the blowing wind , both gradients may be present and the path of the sound waves could be refracted / bend in multiple ways , causing a distant observer to hear the sounds with remarkable clarity.
(b) The wind velocity near the ground is assumed to be horizontal with a vertical gradient v = vx = ky². Here the medium is considered to consist of horizontal layers with different sound velocities. Now , if θ is the angle between the direction of sound wave in the layer and the vertical , and V is the velocity of sound with respect to the ground , then by the law of refraction , sin θ / V = k ( a constant ).
θ1 = θ1 , V 2 = vx + vx sin θ = vx+ v sin θ .
θ2 = θ + d θ1 , V2 = vx + ( v + d v ) sin ( θ + d θ )
Now , using the law of refraction -
[ vx + ( v + dv ) sin ( θ + d θ ) ] / [ vx + v sin θ ] = [ sin θ + d θ ] / sin θ . Now , sin ( θ + d θ ) ≈ sin θ + cos θ d θ , and by retaining the lowest order terms , we get -
dv / vx = d sin θ / sin² θ and so ∫h0 2 ky ( dy / vx) = ∫π/2θ0 ( d sin θ / sin² θ )
[ Note - h is the max height of the parabolic wave front from the horizontal ]
= k h² / vx = 1 / sin θ0 – 1 . Now , the given sound path gives cot θ = dy / dx = ( π h /s ) cos ( π x / s ) or
1 / sin θ = √ ( 1 + cot² θ ) = √ ( 1 + [ π h /s ]² cos² [ π x /s ] )
and 1 / sin θ0 = √ ( 1 + [ π h /s ]² ) . Using this equation provides the path length s , down wind from the sound source where the maximum enhancement of sound intensity occurs as -
s = π vs / √ ( k [ 2 vs + kh² ] ).
(c) Now , the speed of sound in a gaseous medium changes with absolute temperature T as √ T . In the case of a lake surface , it could happen , that a temperature gradient is formed some distance above the surface due to the effect of water ( evaporation ). [ Note that in the problem it's clearly inferred of a windless situation - this is necessary to maintain the stability of a temperature gradient ] . In such a case , sound waves will get refracted resulting in the clarity of sound heard by an observer on the lake shore .