Shortcut Methods
Standing Waves in a Pipe
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Shortcut method: For the fundamental frequency (n = 1), remember the formula: $$f_1 = v/2L$$ where f_1 is the fundamental frequency, v is the speed of sound, and L is the length of the pipe.
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Trick: For overtones, simply multiply the fundamental frequency (f_1) by the overtone number (n). $$f_n = n \times f_1$$
Phenomena of Beats
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Shortcut method: To find the beat frequency, simply take the absolute difference between the frequencies of the two waves. $$f_b = |f_1 - f_2|$$
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Trick: Remember that beats are caused by the interference of two waves with slightly different frequencies. The beat frequency represents the rate at which the waves interfere constructively and destructively, creating the pulsating effect.
Doppler Effect
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Shortcut method:
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For the Doppler shift in frequency, use the formula: $$f_o - f_s = (v/v_s) \times f_s$$ where f_o is the observed frequency, f_s is the source frequency, v is the speed of sound, and v_s is the speed of the source.
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Trick: Remember that the Doppler effect occurs when the source of a sound wave is moving relative to the observer. The shift in frequency depends on the relative velocity between the source and the observer.
