2.1 Sound intensity
For a single source of sound power (watts) the dependence of sound intensity magnitude ( ) on distance (m) from a source is expressed as
The way in which sounds from different sources are added depends on whether or not there is a phase relationship between them. There will be a phase relationship between two loudspeakers connected to the same amplifier. A stereo system will sound best if the loudspeakers are in phase. The loudspeaker sources are said to be coherent sources. Between such sources there can be reinforcement or cancellation depending on position. Usually there is no phase relationship between two separate items of industrial equipment. Such sources are called incoherent . For two such incoherent sources and the combined sound intensity magnitude ( ) at a specific point is given by the sum of the magnitudes of the intensities due to each source at that point. So
where and are the respective sound powers of the sources; and are the respective distances from the point of interest. Note that sound intensity is directional. So if and are on opposite sides of the receiver’s position their intensity contributions will have opposite directions.
Problem in words
With reference to the situation shown in Figure 11, given incoherent point sources and , with sound powers 1.9 W and 4.1 W respectively, 6 m apart, find the sound intensity magnitude at points and at distances and from the line joining and and find the locations of and that correspond to sound intensity magnitudes of 0.02, 0.06 and 0.015 respectively.
Mathematical statement of problem
- Write down an expression for the sound intensity magnitudes at point due to the independent sources A and B with powers and , taking advantage of the symmetry of their locations about the line through at right-angles to the line joining and .
- Find the expression for in terms of and .
- If = 1.9 W, = 4.1 W and = 6m calculate the distance at which the sound intensity is ?
- Find an expression for the intensity magnitude at point .
- Find the value for such that the intensity magnitude at is and the other values are as in part (c).
- Find an equation in powers of relating (intensity magnitude at point ) and .
- By plotting this function for , = 6 m, = 1.9 W, = 4.1 W, find the corresponding values for .
The combined sound intensity magnitude
is given by the sum of the intensity magnitudes due to each source at
. Because of symmetry of the position of
with respect to
Using Pythagoras’ theorem,
the subject of the last formula,
The result that there are two possible values of is a consequence of the symmetry of the sound field about the line joining the two sources. The positive value gives the required location of above the line joining and in Figure 11. The negative value gives a symmetrical location ‘below’ the line.
Note also that if or , then , i.e. would be on the line joining and .
- Using the given values, m.
Using Pythagoras’ theorem again, the distance from
is given by
Multiplying through by
and collecting together like powers of
produces a quartic equation,
Since the quartic equation contains only even powers of , it can be regarded as a quadratic equation in and this can be solved by the standard formula. Hence
Using the given values,
Since must be real, the negative result can be ignored. Hence m.
Using the same procedure as in (d) and (e),
A general expression for the distance at which the intensity at point is is given by collecting like powers of and is another quartic equation, i.e.
Unfortunately this cannot be treated simply as a quadratic equation in since there are terms in odd powers of . One way forward is to plot the curve corresponding to the equation after substituting the given values, another is to use a numerical method such as Newton-Raphson.
Substitution of the given values produces the equation
The plot of the quartic equation in Figure 12 shows that there are two roots of interest. Use of a numerical method for finding the roots of polynomials gives values of the roots to any desired accuracy i.e. m and m.
The result for part (g) implies that there are two locations for along the line joining the two sources where the intensity magnitude will have the given value. One position is about 3.6 m to the left of source and the other is about 4.9 m to the right of source .