what happens if waves that have different properties are combined, and in this section,
we consider such a situation.
To be specific, we consider two traveling waves with the same amplitude but slightly
different frequencies, using sound waves as our example. We examine the waves and
their combination at a fixed x position. You see the graphs above of two such waves
over time at a particular position. Note that in this case the graphs that you see are
displaying the displacement over time of a particle at a fixed position in a medium
carrying longitudinal waves.
The waves were created by tuning forks, and the combined wave is shown below them.
When the waves combine, they produce a wave whose amplitude is not constant, but
instead varies in a repetitive pattern. For sound waves, these “beats” are heard as a
repeating pattern of variation in loudness in a wave of constant frequency.
Musicians sometimes use beats to tune their instruments. Sounds that are close in
frequency produce audible beats, but the beats disappear when the frequencies are the
same. A guitar player, for example, might tune the “A” string by playing an “A” on
another string at the same time and adjusting the tension of the “A” string until there are
no longer any audible beats. This occurs when the frequencies match exactly, or are
close enough that the beats are so far apart in time they can no longer be heard.
We hear beats because the waves constructively and destructively interfere over time.
When they constructively interfere, there is greater condensation and rarefaction of the
air at our ears and we hear louder sounds. Destructive interference means a smaller
change in pressure and a softer sound.
The beat frequency equals the number of times per second we hear a cycle of loud and
soft. This is computed as shown in Equation 1, as the difference of the original
frequencies. When the frequencies are the same, the beat frequency is zero, as when
two strings are perfectly in tune. Humans can hear beats in sound waves at frequencies
up to around 20 beats per second. Above that frequency, the beats are not
distinguishable.
Beat frequency
ƒbeat = ƒ 1 í ƒ 2
ƒbeat = beat frequency
ƒ 1 = frequency of sound one
ƒ 2 = frequency of sound two
(ƒ 1 > ƒ 2 )
What is the beat frequency when
these two waves combine?
ƒ = Ȧ/2ʌ
ƒ 1 = (3380 rad/s)/2ʌ = 538 Hz
ƒ 2 = (3330 rad/s)/2ʌ = 530 Hz
ƒbeat = ƒ 1 íƒ 2
ƒbeat = 538 Hz í 530 Hz
ƒbeat = 8 Hz
17.9 - Gotchas
A standing wave has its maximum displacement at an antinode. Yes, that is correct. An antinode is the opposite of a node, where no motion
occurs.
The locations where peaks and troughs occur are constant in a standing wave. That is correct, and this is the distinguishing point between a
standing wave and a traveling wave.
I see a standing wave on a string with two fixed ends and a single antinode. The wave has two nodes. Yes. The two fixed ends are nodes.
Two waves traveling in the same direction can cause a standing wave. No. Waves traveling in opposite directions can cause a standing wave.
However, a wave from a single source when it is reflected can cause a standing wave, because the reflected wave is traveling in the opposite
direction.