A powerpoint explaining the interference of spherical waves and the physics of beats.

2. • Spherical waves are three dimensional waves.
• Spherical waves oscillate in space and time. However,
their amplitudes remain constant over any spherical
surface centered on the source.
• This allows us to write the wave function of a spherical
wave as
S(r, t) = sm (r)cos (kr - ωt + ϕ)

3. • Locations where two waves are perfectly in phase
• Determined to occur whenever the path difference is
an integer multiple of the wavelength
• Resultant waves can be determined using the formula
S(d, t)= 2sm(d)cos (kd -ωt + ϕ)

4. • Locations where two waves are perfectly out of phase
• Determined to occur when one path is an integer
multiple of wavelengths and the other is a half integer
multiple
• In other words, whenever
Δd= d2 – d1= (n+ 0.5) λ

5. • Two waves with slightly different frequencies have
variation of amplitude which results in a beat.
• When the frequency difference between two sound
waves is very large then we hear two distinct tones
rather than one that varies in intensity
• To determine the resultant wave the equation
Stotal(0, t)=2smcos(ϖt)cos(∆ωt)
can be used with the following quantities
and

6. • The following slide is a video using piano notes to
explain the concepts of consonance, dissonance, beats,
and interference
• Review: All musical notes have their own unique
frequencies.

7. • Question 1:
Two piano keys produce the frequencies of 262
Hz (C) and 330 Hz (G). What is the beat
frequency?

8. • Answer:
330 Hz- 262 Hz= 68 Hz

9. • Question 2:
Why don't we hear beats when different keys
on the piano are played at the same time?

10. • Answer:
In order to hear beats, two interfering sound
waves must have a difference in frequency of 7
Hz or less. No two keys on the piano produce
such a frequency.

11. • Question 3:
If a tuning fork with a frequency of 300 Hz is
played simultaneously with a note with a
frequency of 294 Hz (D). How many beats will
be heard over a period of 10 seconds?

12. • Answer:
300 Hz- 294 Hz= 6 Hz
In 10 seconds, there will be 60 beats.

13. Hawkes Et Al. Physics for Scientists and
Engineers: An Interactive Approach. Vol. 1.
Vancouver: U of British Columbia, n.d. Print.
Thank you for watching!