A string with both ends fixed vibrates in the third harmonic...

A string with both ends fixed vibrates in the third harmonic mode, as shown in Figure $6.18 .$ The length of the string is $6.0 \mathrm{m}$ and the speed of the wave is $120 \mathrm{m} \mathrm{s}^{-1}$ (a) Calculate the wavelength of the wave on the string. (b) The amplitude of oscillation of point $P$ is $4.0 \mathrm{mm} .$ Explain why the displacement of point $P$ is given by the equation $y=4.0 \cos (60 \pi t),$ where $y$ is in millimetres and $t$ is in seconds. (c) The amplitude of oscillation of points $\mathrm{Q}$ and $\mathrm{R}$ is $2.0 \mathrm{mm} .$ State the equation giving the displacement of (i) point $Q$ and (ii) point $R$ (d) Calculate the average speed of (i) point $P$ and (ii) point $Q$ from to $t=0$ to $t=\frac{T}{4}$ where $T$ is the period of the wave. (e) Calculate the maximum speed of (i) point $P$ and (ii) point $Q$

A string with both ends fixed vibrates in the third harmonic mode, as shown in Figure $6.18 .$ The length of the string is $6.0 \mathrm{m}$ and the speed of the wave is $120 \mathrm{m} \mathrm{s}^{-1}$
(a) Calculate the wavelength of the wave on the string.
(b) The amplitude of oscillation of point $P$ is
$4.0 \mathrm{mm} .$ Explain why the displacement of point $P$ is given by the equation $y=4.0 \cos (60 \pi t),$ where $y$ is in millimetres and $t$ is in seconds.
(c) The amplitude of oscillation of points $\mathrm{Q}$ and $\mathrm{R}$ is $2.0 \mathrm{mm} .$ State the equation giving the displacement of (i) point $Q$ and
(ii) point $R$
(d) Calculate the average speed of (i) point $P$ and (ii) point $Q$ from to $t=0$ to $t=\frac{T}{4}$ where $T$ is the period of the wave.
(e) Calculate the maximum speed of (i) point
$P$ and (ii) point $Q$

Key Concepts

Wave Speed

Wave speed is the rate at which a wave propagates through a medium and is related to both the wavelength and frequency of the wave by the equation v = f?. In the context of standing waves, while the overall pattern (nodes and antinodes) does not move, the individual particles of the medium oscillate with speeds determined by the wave’s properties.

Average and Maximum Speeds in Oscillatory Motion

The average speed of a point during a portion of its oscillatory motion can be determined by considering the displacement over a given time interval, while the maximum speed is achieved when the point passes through its equilibrium position. The maximum speed is derived from the amplitude and angular frequency, reflecting how quickly the particle oscillates at its fastest point during the cycle.

Angular Frequency and Period

Angular frequency is related to the temporal frequency of the wave and represents the rate of change of the phase of the sinusoidal function, typically measured in radians per second. The period is the time required for one complete oscillation and is the reciprocal of the frequency. These concepts allow for the characterization of the time dependence in the oscillatory motion of points on the string.

Displacement Equations for Oscillatory Motion

Displacement equations, often expressed in cosine or sine functions, describe the time-dependent displacement of points on a vibrating string. The cosine term indicates the harmonic oscillation with a specific amplitude and angular frequency, and these equations model the periodic motion of points along the string as they oscillate around their equilibrium positions.

Wavelength in Standing Waves

In the context of standing waves on a string fixed at both ends, the wavelength is determined by the resonance condition, where an integer number of half-wavelengths fits exactly within the length of the string. This relationship allows one to calculate the wavelength for any harmonic mode once the string length and harmonic number are known.

Harmonic Modes

Harmonic modes are specific patterns of standing waves that fit an integral number of half-wavelengths into the length of the medium. The fundamental mode has one half-wavelength, while higher harmonics contain more half-wavelengths, leading to more nodes and antinodes. Each successive harmonic represents an increase in frequency and energy of the system.

Standing Waves

Standing waves are the superposition of two traveling waves moving in opposite directions with the same frequency and amplitude, resulting in a pattern that appears to be stationary. They occur in systems with fixed boundaries, such as a string with both ends fixed, where nodes (points of no displacement) and antinodes (points of maximum displacement) are established at specific positions along the medium.

Amplitude

Amplitude is the maximum displacement of a point on the medium from its equilibrium position during oscillation. It represents the energy carried by the wave at that point and varies along the string in a standing wave, being highest at antinodes and zero at nodes.

Transcript

00:01 In this question, we have a string that's in its third harmonic.

00:09 So it will vibrate looking something like that.

00:20 Let me clean that last bit up, just a little bit there for us.

00:25 There are three marked points, p, q, and r.

00:40 We're told that the length of this string is six meters, and the velocity of the wave in the string is 120 meters per second.

00:59 So first, we want to calculate the wavelength of the wave on this string.

01:03 That is pretty simple.

01:06 That's just the wavelength will equal two times the length divided by which harmonic we're in.

01:13 This is the third harmonic.

01:16 So that's two times six divided by three gives us a wavelength of four meters.

01:31 Next, we're told that the amplitude of oscillation at point p is 4 millimeters.

01:40 So i'll say the amplitude at p equals 4 millimeters.

01:48 And we want to explain an equation here.

01:54 An equation for the position or the vertical displacement of p is 4 times the cosine of 60.

02:08 Pi times t so this equation ends up being somewhat simple to explain this four is just the amplitude of the oscillation this 60t is what's called the angular frequency and the angular frequency omega is just two pi times the actual frequency so if we took our actual frequency which we could find from our velocity of a wave equation, velocity of the wave equals the frequency times the wavelength.

02:47 The frequency would equal the velocity divided by the wavelength.

02:53 We would find the frequency is just 30 hertz.

03:03 We then see that 2 pi times 30 hertz gives us that 60 pi for omega that's inside that equation and that just gets multiplied by the time telling you how fast this string cycles.

03:21 The next part of this question says that at points q and r we have an amplitude of two millimeters now and they would like us to write the equation for each of these points.

03:37 It ends up being very similar in each case.

03:44 For the case of q, we have y of q equals now 2 millimeters times the cosine of we have the cosine of we have the same angular frequency, is 60 pi times t.

04:00 But in this case, we're pi out of phase.

04:03 So i could have just put a negative sign out front, but i'll add the pie inside the cosine.

04:10 So a pretty simple equation...

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