# interference and beats

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Interference and Beat Application to Piano

Post on 17-Aug-2015

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Interference and Beat

Application to Piano

• 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)

SPHERICAL WAVES

CONSTRUCTIVE INTERFERENCE

• 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)

• 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

DESTRUCTIVE INTERFERENCE

• 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

BEATS

• 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.

BEATS AND PIANO NOTES

• Question 1:

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

BEATS AND PIANO NOTES

330 Hz- 262 Hz= 68 Hz

BEATS AND PIANO NOTES

• Question 2:

Why don't we hear beats when different keys on the piano are played at the same time?

BEATS AND PIANO NOTES

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.

BEATS AND PIANO NOTES

• 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?

BEATS AND PIANO NOTES