Model The wave is a traveling wave on a stretched string 20 1 Solve The wave speed on a stretched string with linear density is v string T S The wave speed if the tension is doubled will be v string 2 v string 2 200 m s 283 m s T 2 S 20 2 Model The wave is a traveling wave on a stretched string Solve The wave speed on a stretched string with linear density is v string 150 m s 3 333 10 kg m 3 T S 75 N For a wave speed of 180 m s the required tension will be T S v 2 string 3 333 10 kg m 180 m s 3 2 110 N Model The wave pulse is a traveling wave on a stretched string 20 3 Solve The wave speed on a stretched string with linear density is v string T S T S m L LT S m 2 0 m 3 50 10 s 2 0 m 20 N m m 0 025 kg 25 g Model This is a wave traveling at constant speed The pulse moves 1 m to the right every second 20 4 Visualize The snapshot graph shows the wave at all points on the x axis at t 0 s The wave is just reaching x 5 0 The first part of the wave causes an upward displacement of the medium The rising portion of the wave is 2 m wide so it will take 2 s to pass the x 5 0 m point The constant part of the wave whose width is 2 m will take 2 seconds to pass x 5 0 m and during this time the displacement of the medium will be a constant y 1 cm The trailing edge of the pulse arrives at t 4 s at x 5 0 m The displacement now becomes zero and stays zero for all later times Model This is a wave traveling at constant speed The pulse moves 1 m to the left every second 20 5 Visualize This snapshot graph shows the wave at all points on the x axis at t 2 s You can see that the leading edge of the wave at t 2 s is precisely at x 0 m That is in the first 2 seconds the displacement is zero at x 0 m The first part of the wave causes a downward displacement of the medium so immediately after t 2 s the displacement at x 0 m will be negative The negative portion of the wave pulse is 3 m wide and takes 3 s to pass x 0 m The positive portion begins to pass through x 0 m at t 5 s and until t 8 s the displacement of the medium is positive The displacement at x 0 m returns to zero at t 8 s and remains zero for all later times Model This is a wave traveling at constant speed to the right at 1 m s 20 6 Visualize This is the history graph of a wave at x 0 m The graph shows that the x 0 m point of the medium first sees the negative portion of the pulse wave at t 1 0 s Thus the snapshot graph of this wave at t 1 0 s must have the leading negative portion of the wave at x 0 m Model This is a wave traveling at constant speed to the left at 1 m s 20 7 Visualize This is the history graph of a wave at x 2 m Because the wave is moving to the left at 1 m s the wave passes the x 2 m position a distance of 1 m in 1 s Because the flat part of the history graph takes 2 s to pass the x 2 m position its width is 2 m Similarly the width of the linearly increasing part of the history graph is 2 m The center of the flat part of the history graph corresponds to both t 0 s and x 2 m 20 8 Visualize Figure EX20 8 shows a snapshot graph at t 0 s of a longitudinal wave This diagram shows a row of particles with an inter particle separation of 1 0 cm at equilibrium Because the longitudinal wave has a positive amplitude of 0 5 cm between x 3 cm and x 8 cm the particles at x 3 4 5 6 7 and 8 cm are displaced to the right by 0 5 cm 20 9 Visualize We first draw the particles of the medium in the equilibrium positions with an inter particle spacing of 1 0 cm Just underneath the positions of the particles as a longitudinal wave is passing through are shown at time t 0 s It is clear that relative to the equilibrium the particle positions are displaced negatively on the left side and positively on the right side For example the particles at x 0 cm and x 1 cm are at equilibrium the particle at x 2 cm is displaced left by 0 5 cm the particle at x 3 cm is displaced left by 1 0 cm the particle at x 4 cm is displaced left by 0 5 cm and the particle at x 5 cm is undisplaced The behavior of particles for x 5 cm is opposite of that for x 5 cm 20 10 Solve a The wave number is k 2 2 2 0 m 3 1 rad m v f 2 2 0 m 30 rad s 2 9 5 m s b The wave speed is 20 11 Solve a The wavelength is b The frequency is 2 k 2 1 5 rad m 4 2 m f v 200 m s 4 19 m 48 Hz 20 12 Model The wave is a traveling wave Solve 124 rad s and The frequency is 0 rad 0 a A comparison of the wave equation with Equation 20 14 yields A 3 5 cm k 2 7 rad m f 2 124 rad s 2 19 7 Hz 20 Hz 2 k 2 2 7 rad m 2 33 m 2 3 m b The wavelength is c The wave speed v f 46 m s 20 13 Model The wave is a traveling wave Solve rad s and The frequency is 0 rad 0 a A comparison of the wave equation with Equation 20 14 yields A 5 2 cm k 5 5 rad m 72 f 2 72 rad s 2 11 5 Hz 11 Hz 2 k 2 5 5 rad m 1 14 m 1 1 m b The wavelength is c The wave speed v f 13 m s Solve The amplitude of the wave is the maximum displacement which is 6 0 cm The period of the 20 14 wave is 0 60 s so the frequency f T 1 1 0 60 s 1 67 Hz The wavelength is v f 2 m s 1 667 Hz 1 2 m 20 15 …
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