Standing Waves in a Pipe#

Resonant frequencies#

• Formula: $$f_n = nv/2L$$
• (n) is an integer (1, 2, 3, …).
• (v) is the speed of sound in the pipe.
• (L) is the length of the pipe.

Overtones#

• The first overtone is twice the fundamental frequency.
• The second overtone is three times the fundamental frequency, and so on.

Beats#

• Formula: $$f_b = |f_1 - f_2|$$
• (f_b) is the beat frequency.
• (f_1) is the frequency of the first wave.
• (f_2) is the frequency of the second wave.

Doppler effect (Moving source)#

• Formula: $$f’ = f\left(\frac{v \pm v_s}{v \pm v_o}\right)$$
• (f’) is the observed frequency.
• (f) is the source frequency.
• (v) is the speed of sound in the medium.
• (v_s) is the speed of the source.
• (v_o) is the speed of the observer.
• The plus sign is used when the source and observer are moving towards each other and the minus sign is used when they are moving away from each other.

Doppler Effect (Moving observer)#

• Formula: $$f’ = f\left(\frac{v \pm v_o}{v \mp v_s}\right)$$
• (f’) is the observed frequency.
• (f) is the source frequency.
• (v) is the speed of sound in the medium.
• (v_s) is the speed of the source.
• (v_o) is the speed of the observer.
• The plus sign is used when the observer is moving towards the source and the minus sign is used when they are moving away from the source.