Determine the combined intensity of two waves: Perfect constructive interference

We know from Superposition and Interference that when two identical waves, which have equal amplitudes $X$, interfere perfectly constructively, the resulting wave has an amplitude of $2X$. Because a wave’s intensity is proportional to amplitude squared, the intensity of the resulting wave is four times as great as in the individual waves.

1. Recall that intensity is proportional to amplitude squared.
2. Calculate the new amplitude:
   $$I\prime \propto {\left(X\prime \right)}^2={\left(2X\right)}^2={4X}^2.$$

   16.86
3. Recall that the intensity of the old amplitude was:
   $$I^{}\propto X^2.$$

   16.87
4. Take the ratio of new intensity to the old intensity. This gives:
   $$\frac{I\prime }{I}=4.$$

   16.88
5. Calculate to find $I\prime$:
   $$I\prime =4I=4\text{.}\text{00}\mathrm{<mspace\; width="0.25em"><msup><mtext>W/m</mtext><mrow><mn>2</mn></mrow></msup></mspace>}.$$

   16.89

The intensity goes up by a factor of 4 when the amplitude doubles. This answer is a little disquieting. The two individual waves each have intensities of $1\text{.}\text{00}\mathrm{<mspace\; width="0.25em"><msup><mtext>W/m</mtext><mrow><mn>2</mn></mrow></msup></mspace>}$, yet their sum has an intensity of $4\text{.}\text{00}\mathrm{<mspace\; width="0.25em"><msup><mtext>W/m</mtext><mrow><mn>2</mn></mrow></msup></mspace>}$, which may appear to violate conservation of energy. This violation, of course, cannot happen. What does happen is intriguing. The area over which the intensity is $4\text{.}\text{00}\mathrm{<mspace\; width="0.25em"><msup><mtext>W/m</mtext><mrow><mn>2</mn></mrow></msup></mspace>}$ is much less than the area covered by the two waves before they interfered. There are other areas where the intensity is zero. The addition of waves is not as simple as our first look in Superposition and Interference suggested. We actually get a pattern of both constructive interference and destructive interference whenever two waves are added. For example, if we have two stereo speakers putting out $1\text{.}\text{00}\mathrm{<mspace\; width="0.25em"><msup><mtext>W/m</mtext><mrow><mn>2</mn></mrow></msup></mspace>}$ each, there will be places in the room where the intensity is $4\text{.}\text{00}\mathrm{<mspace\; width="0.25em"><msup><mtext>W/m</mtext><mrow><mn>2</mn></mrow></msup></mspace>}$, other places where the intensity is zero, and others in between. Figure 16.45 shows what this interference might look like. We will pursue interference patterns elsewhere in this text.

Figure 16.45 These stereo speakers produce both constructive interference and destructive interference in the room, a property common to the superposition of all types of waves. The shading is proportional to intensity.