or to represent a simple wave, where is the wave number and the angular frequency. If the sign of is minus, the wave is propagating along the positive direction of x-axis. Vice versa , if the sign of is positive, the wave is propagating along the negative direction of x-axis. is the initial phase.For simplicity of calculation, we also use an equivalent function to represent a simple wave ( , , ) .
The physical quantity can be simply extracted from the real part of the function. When two waves of same amplitude and frequency travel in opposite direction along a rope, standing wave may form by superposition of the two waves under particular conditions. We are going to discuss the conditions of forming a standing wave on a stretched rope with two fixed ends.
Suppose a traverse wave is traveling from x=0 toward the positive direction on a rope of length (see figure 1).
The mathematical form of the wave is given by (1)
is the amplitude. A reflected wave forms when the wave reaches the fixed end at .
Suppose there is no energy adsorption at the point of reflection thus the amplitude and frequency of the reflected wave remains the same as the incident wave. However the phase of the wave may alter. So the reflected wave can be written as
The superposition of the incident and reflected waves is
Using the boundary condition at
we have,and get
Now, we can rewrite equation (3) as
Since the rope is also fixed at implying that.
So , (6)
Equation (6) is the condition for forming a standing wave on a rope with given length of .
Since ,the condition also can be read as which means that the length of the rope is always equal to the half of the wavelength multiplying by an integer.
he wave function of the standing wave is the real part of that is
The value of in equation (7) is the maximum amplitude of the standing wave. The static points on the rope are called nodes. According to equation (7) the nodes happen at positions . While those points oscillate with maximum amplitude are called antinodes. Alphabets N and A in figure 2 indicate the positions of nodes and antinodes.
Does standing wave benefit to the propagation of signal ?
2、The relation between wave speed and tension
Let us consider a pulse wave propagating on a string. Assume that the pulse wave is propagating toward the direction of the right hand side when the string is at rest relative to an observer. If the string length is infinity and the string is moving at the same speed of the pulse wave but in opposite direction, the pulse wave becomes static in space relative to the observer(see figure 3).
Let be the mass per unit length of the string and F is the force that maintains the segment of the string moving in a circular orbit of radius R. The centrifugal force can be approximated by because the angle is infinitely small or .
According to Newton’s second law of motion () , we have . is the mass of the segment of the string. From figure 3 we know the relation . So we have and the propagating speed of the wave can be obtained from the speed of circular motion.
3、The relation between vibration frequency and string tension
Suppose a standing wave forms on a string of length Equation (6) gives . From the relation , we get
According to the equation (9) if the frequency of vibration f and the mass per unit length of the string are both constant then
The ratio of tension and the square of the wave length is a constant. This relation can be indentified in the experiment.