1528 
The spread of the profile makes (the reciprocal of 
the time constant) in the above theory a function of distance of 
propagation. We take this effect roughly into account in the 
ultimate numerical calculations by using an average value of 
the time constant over the region of propagation to determine 
A » but continue to treat ot as a constant in the mathe- 
matical analysis. 
The sharpening effect is more significant in modifying 
the shape of the front of the wave, and we attempt in this 
section to make some correction for it. 
9.2 Assuming that the front of the wave has developed a finite 
rise time, we see (as a result of the theory of finite amplitude 
waves) that the initial portion of the wave will propagate at 
the velocity of sound Co while the peak of the wave will propagate 
approximately at the velocity C+u, where C is the local velocity 
of sound and & the particle velocity. Since (C+u) > Cy ; 
it is evident that the effect of finite amplitude is opposite 
to that of viscous attenuation in that it tends to decrease the 
rise time and sharpen the front. From a slightly different 
point of view, we can regard this approximately'as a process of 
restoration of higher frequency components to the spectrum of 
the pulse. 
An outline of the correction proceedure follows: We 
calculate the rate at which the rise time is increasingwith 
distance from the origin owing to viscous attenuation and also 
the rate at which it is decreasing owing to finite amplitude. 
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