the speed of a wave on a string

The speed of a wave on a string

• One of the key properties of any wave is the wave speed.

• Consider a string in which the tension is F and the linear

mass density (mass per unit length) is μ.

• We expect the speed of transverse waves on the string v

should increase when the tension F increases, but it should

decrease when the mass per unit length μ increases.

• It is shown in your text that the wave speed is:

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The four strings of a musical instrument are all made of the same

material and are under the same tension, but have different

thicknesses. Waves travel

A. fastest on the thickest string.

B. fastest on the thinnest string.

C. at the same speed on all strings.

QuickCheck


Example 1


Example 2 – One end of a 2.00-kg rope is tied to a support at the top of a

mine shaft 80.0 m deep. The rope is stretched taut by a 20.0-kg box of

rocks attached at the bottom. A geologist at the bottom of the shaft

signals to a colleague at the top by jerking the rope sideways. What is

the speed of a transverse wave on the rope? Also, if a point on the rope

is in transverse SHM with f = 2.00 Hz, how many cycles of the wave are

there in the rope’s length?

Reflection of a wave pulse at a fixed end of a string

• What happens when a wave

pulse or a sinusoidal wave

arrives at the end of the string?

• If the end is fastened to a rigid

support, it is a fixed end that

cannot move.

• The arriving wave exerts a

force on the support (drawing

4).


Reflection of a wave pulse at a fixed end of a string

• The reaction to the force of

drawing 4, exerted by the

support on the string, “kicks

back” on the string and sets up a

reflected pulse or wave traveling

in the reverse direction.


Reflection of a wave pulse at a free end of a string

• A free end is one that is

perfectly free to move in the

direction perpendicular to the

length of the string.

• When a wave arrives at this

free end, the ring slides along

the rod, reaching a maximum

displacement, coming

momentarily to rest (drawing

4).


Reflection of a wave pulse at a free end of a string

• In drawing 4, the string is now

stretched, giving increased

tension, so the free end of the

string is pulled back down, and

again a reflected pulse is

produced.


Standing waves on a string

• Waves traveling in opposite directions on a taut string

interfere with each other.

• The result is a standing wave pattern that does not move on

the string.

• Destructive interference occurs where the wave

displacements cancel, and constructive interference occurs

where the displacements add.

• At the nodes no motion occurs, and at the antinodes the

amplitude of the motion is greatest.


Standing Waves

Time snapshots of two sine waves.

The red wave is moving in the −x-

direction and the blue wave is

moving in the +x-direction. The

resulting wave is shown in black.

Consider the resultant wave at the

points x = 0 m, 3 m, 6 m, 9 m, 12

m, 15 m and notice that the resultant

wave always equals zero at these

points, no matter what the time is.

These points are known as fixed

points (nodes).

In between each two nodes is an

antinode, a place where the medium

oscillates with an amplitude equal to

the sum of the amplitudes of the

individual waves.

FIGURE 16.27

𝑦 𝑥, 𝑡 = 2𝐴𝑠𝑖𝑛 𝑘𝑥 cos(𝜔𝑡)

The sine function dictates the position of the standing waves

while the cosine function expresses how the shape

oscillates with time.


What is the wavelength

of this standing wave?

QuickCheck

A. 0.25 m

B. 0.5 m

C. 1.0 m

D. 2.0 m

Slide 17-12


Example 3

Example 4 – A guitar string is plucked and creates a standing sinusoidal

wave with amplitude 0.750 mm and frequency 440 Hz. The wave velocity

is 143 m/s.

(a) Find the equation of the standing wave.

(b) Locate the nodes

(c) Find the maximum speed and acceleration of the string.


• This is a time exposure of a

standing wave on a string.

• This pattern is called the

second harmonic.



• As the frequency of the

oscillation of the right-hand

end increases, the pattern of

the standing wave changes.

• More nodes and antinodes

are present in a higher

frequency standing wave.


Normal modes

• For a taut string fixed at both

ends, the possible wavelengths

are and the possible

frequencies are fn = n v/2L =

nf1, where n = 1, 2, 3, …

• f1 is the fundamental

frequency, f2 is the second

harmonic (first overtone), f3 is

the third harmonic (second

overtone), etc.

• The figure illustrates the first

four harmonics.



What is the mode number

of this standing wave?

QuickCheck

A. 4

B. 5

C. 6

Slide 17-18


QuickCheck

A standing wave on a string vibrates as shown.

Suppose the string tension is reduced to 1/4 its

original value while the frequency and length are

kept unchanged. Which standing wave pattern is

produced?

Slide 17-19

Standing waves and string instruments

• When a string on a musical instrument is plucked, bowed or

struck, a standing wave with the fundamental frequency is

produced:

• This is also the frequency of the sound wave created in the

surrounding air by the vibrating string.

• Increasing the tension F increases the frequency (and the

pitch).


Example 5 - Adjacent antinodes of a standing wave on a string are 15.0 cm

apart. A particle at an antinode oscillates in simple harmonic motion with

amplitude 0.850 cm and period 0.0750 s. The string lies along the +x-axis and

is fixed at x = 0.

(a) How far apart are the adjacent nodes? How far from antinodes?

(b) Find the wavelength, amplitude, and speed of the standing wave.

(c) Find the wavelength, amplitude, and speed of the traveling waves.

(d) Find the max speed of the string.

In-class Activity #1 – A standing wave on a wire has an amplitude of 2.40

mm, an angular frequency of 934 rad/s, and wave number 0.750π rad/m.

The left end of the wire is at x = 0. At what distances from the left end are

(a) the nodes of the standing wave?

(b) the antinodes of the standing wave?

(c) What is the node to antinode distance?

Organ pipes

• Organ pipes of different sizes produce tones

with different frequencies (bottom figure).

• The figure at the right shows displacement

nodes in two cross-sections of an organ pipe

at two instants that are one-half period

apart. The blue shading shows pressure

variation.

© 2016 Pearson Education Inc.

Harmonics in an open pipe

• The fundamental frequency of an open pipe is shown.

• The shading indicates the pressure variations.

• The red curves are graphs of the displacement along the pipe

axis at two instants separated in time by one half-period.


Harmonics in an open pipe

• Higher harmonics in an open pipe have frequency:


Harmonics in a stopped pipe

• The fundamental frequency of a stopped pipe is shown.

• The shading indicates the pressure variations.

• The red curves are graphs of the displacement along the pipe

axis at two instants separated in time by one half-period.


Harmonics in a stopped pipe

• Higher harmonics in a stopped pipe have frequency:



QuickCheck

An open-open tube of air has

length L. Which is the

displacement graph of the m = 3

standing wave in this tube?

Slide 17-30


QuickCheck

An open-closed tube of air of length

L has the closed end on the right.

Which is the displacement graph of

the m = 3 standing wave in this tube?

Slide 17-31


Example 6 – One a day when the speed of sounds is 345

m/s, the fundamental frequency of a particular stopped-organ

pipe is 220 Hz. How long is this pipe? The second overtone

of this pipe has the same wavelength as the third harmonic of

an open pipe. How long is the open pipe?

the speed of a wave on a string

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