弦线驻波

 Experiment 39: String standing wave

1 Experimental purpose

Observe the standing wave phenomenon formed on strings and rings, study the conditions and rules for the generation of standing waves, and understand the normality of string vibration.

Patterns and sound patterns of stringed instruments.

2. Experimental device

The experimental device is shown in Figure 39-1, including an electric oscillator, a vibration source, a string (circular elastic band), and a steel wire ring.

root. The vibration signal is generated by the electric oscillator and output to the vibration source in the vibration source box. The vibration source is modified from a speaker and connected to

A vibrating rod, the vibration source drives the vibrating rod to vibrate up and down. The vibration frequency and amplitude of the vibrating rod can be determined by the electric oscillator

The frequency adjustment knob and power adjustment knob control and adjust. The vibrating rod can drive the string or steel wire fixed on the rod to vibrate.

move.

1

2

3 4

5

1 6

2

3 4

5

6

Figure 39-1

In the picture: 1-electric oscillator, 2-vibration source box, 3-vibration rod, 4-fixed rod, 5-amplitude position, 6-node position

3 Experimental steps and methods

1 Generation of standing waves in strings

As shown in Figure 39-1, fix both ends of the string to the fixed rod on the electric oscillator box and the vibrating rod on the vibration source box.

superior. Adjust the frequency adjustment knob of the power supply to the lowest position and the power adjustment knob to the middle position, and connect the output of the electric oscillator.

The output terminal goes to the input terminal of the vibration source box.

Turn on the power and gradually turn the frequency adjustment knob from low to high. You can observe that one to several standing waves will appear on the string.

tricks.

2 Generation of ring standing waves

As shown in Figure 39-2, fix the wire ring on the vibrating rod. Turn on the power and adjust the frequency knob and power knob from small to large.

button, several ring-shaped standing waves will be formed on the ring.

1

2

3 4

6

5

1

2

3 4

6

5

Figure 39-2

In the picture: 5-wire ring, 6-standing wave shape on the wire ring

Four Experimental Principles

Suppose two columns of simple harmonics with the same frequency, same vibration direction, and equal amplitude propagate in opposite directions on the same straight line,

The wave equation is

⎩

⎨

⎧

ϕ++ω=

ϕ+−ω=

)kxtcos(Ay

)kxtcos(Ay

twenty two

1 1 (1)

In the formula, A is the amplitude of the wave, is the circular frequency, ω = π /2k λ , λ is the wavelength, ϕ1 and ϕ2 are the initial phases of the two vibrations respectively.

The resultant motion produced by the superposition of the two waves is

) 2 tcos() 2 kxcos(A2yyy 12 12

twenty one

ϕ+ϕ +ω ϕ−ϕ =+= + (2)

This is the standing wave equation.

It can be seen from the standing wave equation that when a standing wave is formed, each point on the straight line performs simple harmonic motion near its equilibrium position.

Some points have always the largest amplitude due to superposition construction (the maximum amplitude is 2A), which are called antinodes. The conditions that the antinode satisfies are

...3,2,1n,n 2

kx 12 =π= ϕ−ϕ + , the distance between two adjacent antinodes is λ 2/ . Some points begin due to superposition and cancellation

The end is still and does not move (the minimum amplitude is zero), which is called a node. The conditions satisfied by the wave node are

...3,2,1n, 2

)1n2( 2

kx 12 = π

+= ϕ−ϕ + , the distance between two adjacent wave nodes is also λ 2/ .

It is easy to deduce from the characteristics of standing waves that not waves of any wavelength or frequency can form standing waves on a string of a certain length.

of. For a string with fixed ends, when a standing wave is formed, the two ends of the string are nodes. At this time, the wavelength and string length l

The following relationships are satisfied:

λ

⋅⋅⋅= λ = 3,2,1n 2

nl (3)

That is, the string length l is equal to an integer multiple of half the wavelength. According to the wave speed v = λν , the frequency of the string standing wave should satisfy ν

...3,2,1n l2

v

n

v n = = λ

=ν (4)

Each frequency corresponds to a possible vibration mode of the string. These frequencies are called the eigenfrequencies of the string vibration, also known as the abbreviation

positive frequency. The various vibration modes determined by equation (3) or (4) are called the normal modes of string vibration. When the outside world instigates

When the frequency of the source is the same as a certain normal frequency of the string, a strong standing wave will be excited on the string due to resonance.

In the string standing wave experiment, the vibrating rod drives one end of the string to vibrate, producing a row of traveling waves in the string with a wave speed of

μ

F

v = (5)

In the formula, F is the tension on the string, and μ is the mass linear density of the string (the mass of the string per unit length). The traveling wave passes through another string

After reflection at the fixed end, two columns of traveling waves traveling toward each other are superimposed on the string line, forming a standing wave on the string line. Adjust the incident wave frequency,

When the conditions of equation (3) or (4) are met, n standing waves with antinodes can be formed on the string.

In the ring standing wave experiment, the vibrations from the left and right ends of the fixed point of the steel wire are superimposed on the steel wire. When adjusted to the circumference of the circle

When equal to n (integer) times of half wavelength, a ring-shaped standing wave with n antinodes can be formed on the steel wire.

Five things to note

1. During the experiment, the output power of the electric oscillator should not be adjusted too high to avoid causing severe vibration with a large amplitude and damaging the device.

Bringing potential safety hazards (such as vibration breakage of wire rings, etc.).

2. When adjusting the frequency knob, be careful to adjust it slowly to meet the frequency conditions for standing wave generation and form a stable standing wave.

Six questions

1. In the string standing wave experiment, if the frequency is fixed and the length of the string is changed, can various standing wave patterns be formed? Observe through experiments

And verification.

2 Stringed instruments such as violins, erhus, and guitars all rely on the vibration of strings to produce sound. When playing, draw with a bow or use your hands

By plucking the string with your finger, you can force the string to vibrate; by pressing somewhere on the string with your finger, you can change the length of the vibrating part of the string.

and emit musical tones of different pitches. The eigenfrequency of the string standing wave satisfies formula (4), where when n=1, the corresponding frequency

for l2

v

1 =ν, this frequency is called the fundamental frequency, which determines the pitch of the string vibration; when n=2, 3,..., the corresponding frequency

are , respectively called the second, third... harmonics, which determine the timbre of the string vibration. 1n nν=ν

Please draw the standing wave pattern generated on a string with both ends fixed when n=1, n=2, n=3.

3. When you blow air into an empty Sprite bottle with the side of your mouth facing the mouth of the bottle, you can hear the sound caused by the vibration of the air in the bottle. Various

Wind instruments produce sound by vibrating the air inside the tube. Figure 39-3 depicts the inner space of a tube with one end open and one end closed.

The fundamental frequency, second harmonic frequency, third harmonic frequency and other normal modes when gas vibration forms a standing wave, in which the closed end is the node and the open end is

antinode. Similar to string vibration, the fundamental frequency determines the pitch of the tube vibration, and the harmonic frequency determines the timbre of the tube vibration.

Assume the height of the pipe column is L and the sound speed is V. Find the wavelengths and frequencies corresponding to the normal modes of various pipe vibrations in the figure.

L

1 2 3

Figure 39-3

In the figure: 1-Fundamental frequency mode, 2-Second harmonic mode, 3-Third harmonic mode

4 Figure 39-4 shows the staff of Lao She Teahouse in Beijing performing "Wan Qin". Wanqin is a typical representative of Dawancha culture.

Using 32 tea bowls of different sizes, music of different pitches can be struck.

Please use the objects or materials you can find around you (such as bottles, basins, cups,

Nutshells, metal sheets, strings, etc.), design and make a simple musical instrument, and after self-practice, play it for everyone to enjoy.