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127. Problem 19.27 (RHK) A wave travels out uniformly in all directions from a point source. (a) We have to justify that the expression for the displacement y of the medium at any distance r from the from the source is
We have to consider the speed, direction of propagation, periodicity, and intensity of the wave. (b) We have to find the dimension of the constant Y. |
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Solution: Click For PDF Version (a) The intensity, I, of a spherical wave at a distance r from its source
has to vary as
We will justify that the function
describes spherical waves. As the function
(b) As y represents the displacement of the medium, its dimension has to
be that of length
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, because
, which
is the total amount of energy flowing per second through a spherical surface of
radius r centred at the source, is equal to the power of the source and
is therefore a number independent of the variable r . We know that the
intensity of a wave is proportional to the modulus square of its amplitude.
is of the generic form
it
represents wave motion along the direction of increasing r. From this
function we note that the speed of the wave is v and the frequency of the
wave
, where
k the wave number of the wave determines the wavelength
.
As the intensity of wave motion is proportional to the square of the amplitude,
which for the function is
, where
Y is a constant, it describes spherical wave propagation. 
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Therefore, the dimension of the constant Y is
.