ism_chapter_16

# ism_chapter_16 - Chapter 16 Superposition and Standing...

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1221 Chapter 16 Superposition and Standing Waves Conceptual Problems *1 •• Picture the Problem We can use the speeds of the pulses to determine their positions at the given times. 2 •• Picture the Problem We can use the speeds of the pulses to determine their positions at the given times. 3 Determine the Concept Beats are a consequence of the alternating constructive and destructive interference of waves due to slightly different frequencies. The amplitudes of the waves play no role in producing the beats. correct. is ) ( c 4 ( a ) True. The harmonics for a string fixed at both ends are integral multiples of the frequency of the fundamental mode (first harmonic). ( b ) True. The harmonics for a string fixed at both ends are integral multiples of the

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Chapter 16 1222 frequency of the fundamental mode (first harmonic). ( c ) True. If l is the length of the pipe and v the speed of sound, the excited harmonics are given by l 4 v n f n = , where n = 1, 3, 5… 5 •• Determine the Concept Standing waves are the consequence of the constructive interference of waves that have the same amplitude and frequency but are traveling in opposite directions. correct. is ) ( b *6 Determine the Concept Our ears and brain find frequencies which are small-integer multiples of one another pleasing when played in combination. In particular, the ear hears frequencies related by a factor of 2 (one octave) as identical. Thus, a violin sounds much more "musical" than the sound of a drum. 7 Picture the Problem The first harmonic displacement-wave pattern in an organ pipe open at both ends and vibrating in its fundamental mode is represented in part ( a ) of the diagram. Part ( b ) of the diagram shows the wave pattern corresponding to the fundamental frequency for a pipe of the same length L that is closed at one end. Letting unprimed quantities refer to the open pipe and primed quantities refer to the closed pipe, we can relate the wavelength and, hence, the frequency of the fundamental modes using v = f λ . Express the frequency of the first harmonic in the open pipe in terms of the speed and wavelength of the waves: 1 1 v f = Relate the length of the open pipe to the wavelength of the fundamental mode: L 2 1 =
Superposition and Standing Waves 1223 Substitute to obtain: L v f 2 1 = Express the frequency of the first harmonic in the closed pipe in terms of the speed and wavelength of the waves: ' v ' f 1 1 λ = Relate the length of the closed pipe to the wavelength of the fundamental mode: L ' 4 1 = Substitute to obtain: 1 1 2 1 2 2 1 4 f L v L v ' f = = = Substitute numerical values and evaluate ' f 1 : () Hz 200 Hz 400 2 1 1 = = ' f and correct. is ) ( a 8 •• Picture the Problem The frequency of the fundamental mode of vibration is directly proportional to the speed of waves on the string and inversely proportional to the wavelength which, in turn, is directly proportional to the length of the string. By expressing the fundamental frequency in terms of the length L of the string and the tension F

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ism_chapter_16 - Chapter 16 Superposition and Standing...

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