Physics for Scientists and Engineers, 6e Chapter 18 – Superposition and Standing Waves

the energy associated with the pulses has disappeared Two pulses move in opposite directions on a string and are identical in shape except that one has positive displacements of the elements of the string and the other has negative displacements. At the moment that the two pulses completely overlap on the string, the energy associated with the pulses has disappeared the string is not moving the string forms a straight line the pulses have vanished and will not reappear 1 2 3 4 5

The pulses completely cancel each other in terms of displacement of elements of the string from equilibrium, but the string is still moving. A short time later, the string will be displaced again and the pulses will have passed each other.

is zero for all elements is positive for all elements Consider a standing wave on a string as shown in the figure below. Define the velocity of elements of the string as positive if they are moving upward in the figure. At the moment the string has the shape shown at the bottom of part (a), the instantaneous velocity of elements along the string is zero for all elements is positive for all elements is negative for all elements varies with the position of the element 1 2 3 4 5

The pattern shown at the bottom of Figure 18 The pattern shown at the bottom of Figure 18.9a corresponds to the extreme position of the string. All elements of the string have momentarily come to rest.

is zero for all elements is positive for all elements Continuing with the scenario in question 3, at the moment the string has the shape shown at the bottom of part b of the figure above, the instantaneous velocity of elements along the string is zero for all elements is positive for all elements is negative for all elements varies with the position of the element 1 2 3 4 5

Near a nodal point, elements on one side of the point are moving upward at this instant and elements on the other side are moving downward.

When a standing wave is set up on a string fixed at both ends, the number of nodes is equal to the number of antinodes the wavelength is equal to the length of the string divided by an integer the frequency is equal to the number of nodes times the fundamental frequency the shape of the string at any time is symmetric about the midpoint of the string. 1 2 3 4 5

Choice (1) is incorrect because the number of nodes is one greater than the number of antinodes. Choice (2) is only true for half of the modes; it is not true for any odd-numbered mode. Choice (3) would be correct if we replace the word nodes with antinodes.

fclosed = fopen fclosed = 1/2 fopen fclosed = 2 fopen A pipe open at both ends resonates at a fundamental frequency fopen. When one end is covered and the pipe is again made to resonate, the fundamental frequency is fclosed. Which of the following expressions describes how these two resonant frequencies compare? fclosed = fopen fclosed = 1/2 fopen fclosed = 2 fopen fclosed = 3/2 fopen 1 2 3 4 5

With both ends open, the pipe has a fundamental frequency given by Equation 18.11: fopen = v / 2L. With one end closed, the pipe has a fundamental frequency given by Equation 18.12:

is impossible to determine Balboa Park in San Diego has an outdoor organ. When the air temperature increases, the fundamental frequency of one of the organ pipes stays the same goes down goes up is impossible to determine 1 2 3 4 5

The increase in temperature causes the speed of sound to go up The increase in temperature causes the speed of sound to go up. According to Equation 18.11, this will result in an increase in the fundamental frequency of a given organ pipe.

continue to tighten the string loosen the string You are tuning a guitar by comparing the sound of the string with that of a standard tuning fork. You notice a beat frequency of 5 Hz when both sounds are present. You tighten the guitar string and the beat frequency rises to 8 Hz. In order to tune the string exactly to the tuning fork, you should continue to tighten the string loosen the string impossible to determine 1 2 3 4 5

Tightening the string has caused the frequencies to be farther apart, based on the increase in the beat frequency.