University of Wisconsin Green Bay

You are listening to your stereo at a location where the intensity of the sound wave is 0.00014 W/m2. What is the intensity of the sound wave if you move twice as far away from the speakers? (Hint: Treat the sound wave as a spherical wave.)

  • In this problem, you are given the intensity of a wave at one distance and asked the intensity of the same wave at another distance. Intensity of a wave is defined as the power per unit area, and area depends on distance from the source, so intensity is closely related to power through distance. This is a definition of intensity problem.

    You might notice at this point that you don’t have enough information (you don’t know distance) to plug directly into the definition equation. If you don’t notice that now, don’t worry about it. It will become apparent in the solution. However, if you do notice it now, it should not stop you from continuing with the solution. A very common approach when you are given information about one point and asked for similar information about another is to take the ratio of the definition equation at the two points. In this way, the unknown information is divided out.

    This definition problem will be solved taking ratios.

  • Intensity of Sound --

    There is no need for a picture in most definition problems, and this is one of them. You are given distance and intensity information at one point and asked for the intensity at another distance. A picture will not provide any additional organization beyond what is already present in the problem. However, if picturing the wave helps to understand the equation, then you should absolutely make any sort of sketch that is helpful to you.

  • In equation form, the intensity of a spherical wave is given by

    I = Power / Area = P / (4πr2)

    This is the only relation you need for this problem.

  • -- --

    I = P / (4πr2)

    If you didn’t recognize it earlier, you now see that you cannot answer the question just by plugging in information about the point of interest (your new location.) But you are given information about two different points, and so you can solve the problem by taking the ratio of this equation at the two points:

    I2 / I1 = [P / (4πr2)]2 / [P / (4πr2)]1

    I2 / I1 = P(4πr12) / P(4πr22)

    I2 / I1 = r12 / (2 r1)2

    I2 / I1 = 1 / 22 = 1 / 4

    I2 = 1 / 4 (I1) = 1 / 4 (0.00014 W/m2)

    I2 = 0.000035 W/m2

    The problem asked for the intensity of the sound wave at your new location (point 2.) No further mathematical solution is necessary.

  • I2 / I1 = [P/(4πr2)]2 / [P/(4πr2)]1
    I2 / I1 = r12 / (2 r1)2
    I2 = 1/4 (I1) = 0.000035 W/m2

    There are two parts to understanding this problem. The first is just the relationship between intensity and distance. For a wave that spreads out spherically, intensity decreases with distance squared and so you expect to find intensity down by a factor of four when you double the distance.

    The second thing to understand in this problem is the usefulness of solving problems with ratios. This type of problem frequently occurs on standardized exams such as the GRE and MCAT. Whenever you are given limited information about two points, and especially if you are asked to compare the two points, it is likely that you can solve the problem most efficiently by taking a ratio of the appropriate equation (usually a definition) at the two points.