3.14.3 Experimental principles
1)The formation of the standing waves
A standing wave is a combination of two simple harmonic waves with the same frequency,the same vibration direction,the same amplitude,and the opposite propagation direction.Assuming that the two waves in Figure 3.14.3 propagate in the opposite directions along the axis.The rightpropagating wave is represented by a thin solid line,while the left-propagating wave is represented by a thin dashed line,and their composite standing wave is represented by a thick solid line.
Figure 3.14.3 The schematic diagram of standing wave formation
Supposing that the wave propagating along the positive direction of the axis is the incident wave,and the wave propagating along the negative direction of the axis is the reflected wave.Their wave equations are:
Where A is the amplitude of the simple harmonic wave;f is the frequency;λ is the wavelength,and x is the coordinate position of the mass point on the string.The composite wave after the superposition of the two waves is a standing wave,and its equation is:
It can be seen that all points on the string are making simple harmonic vibrations at the same frequency,and their amplitude is
,that is,the amplitude of the standing wave has nothing to do with time,but only with the position of the mass point.
If
that is
(k=0,1,2,…),the corresponding amplitude is the maximal 2A,which is called the antinode position of the standing wave.If
,that is x=(2k+1)λ/4(k=0,1,2,…),the corresponding amplitude is the minimum zero,which is called the node position of the standing wave.
It can be seen from the discussion above that the distance between two adjacent antinodes(or nodes)is equal to half wavelengthΔx=λ/2.Therefore,in the standing wave experiment,as long as the distance between two adjacent nodes or two adjacent antinodes is measured,the wavelength of the wave can be determined.In the experiment,the two ends of the fixed string become the nodes of the standing wave.
Therefore,only when the distance between the two fixed ends of the string(chord length)is equal to an integral multiple of the half-wavelength,the standing wave condition can be formed.The mathematical expression is:
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Therefore,the wavelength of the transverse wave propagating along the string is
2)Stri ng vibration law
According to the wave theory,the propagation speed of the transverse wave in the string is:
Where T is the tension of the string and ρ the linear density of the string.According to the relational formula of the wave speed,frequency and wavelength,Equation(3.14.4)can be rewritten into:
Then,according to Equation(3.14.5)and Equation(3.14.6),the following relation can be obtained:
Then we can derive:
According to Equation(3.14.7),when T,ρ,and L are given,a standing wave on the string can only be formed when the frequency f is satisfied.
