How do standing waves happen

As the two pulses pass through each other, they will undergo destructive interference. Thus, a point of no displacement in the exact middle of the snakey will be produced. The animation below shows several snapshots of the meeting of the two pulses at various stages in their interference. The individual pulses are drawn in blue and red; the resulting shape of the medium as found by the principle of superposition is shown in green.

Note that there is a point on the diagram in the exact middle of the medium that never experiences any displacement from the equilibrium position.

An upward displaced pulse introduced at one end will destructively interfere in the exact middle of the snakey with a second upward displaced pulse introduced from the same end if the introduction of the second pulse is performed with perfect timing. The same rationale could be applied to two downward displaced pulses introduced from the same end. If the second pulse is introduced at precisely the moment that the first pulse is reflecting from the fixed end, then destructive interference will occur in the exact middle of the snakey.

The above discussion only explains why two pulses might interfere destructively to produce a point of no displacement in the middle of the snakey. A wave is certainly different than a pulse. What if there are two waves traveling in the medium? Understanding why two waves introduced into a medium with perfect timing might produce a point of displacement in the middle of the medium is a mere extension of the above discussion.

While a pulse is a single disturbance that moves through a medium, a wave is a repeating pattern of crests and troughs. Thus, a wave can be thought of as an upward displaced pulse crest followed by a downward displaced pulse trough followed by an upward displaced pulse crest followed by a downward displaced pulse trough followed by Since the introduction of a crest is followed by the introduction of a trough, every crest and trough will destructively interfere in such a way that the middle of the medium is a point of no displacement.

Of course, this all demands that the timing is perfect. In the above discussion, perfect timing was achieved if every wave crest was introduced into the snakey at the precise time that the previous wave crest began its reflection at the fixed end. In this situation, there will be one complete wavelength within the snakey moving to the right at every instant in time; this incident wave will meet up with one complete wavelength moving to the left at every instant in time.

Under these conditions, destructive interference always occurs in the middle of the snakey. Either a full crest meets a full trough or a half-crest meets a half-trough or a quarter-crest meets a quarter-trough at this point. The animation below represents several snapshots of two waves traveling in opposite directions along the same medium.

Nodes appear at integer multiples of half wavelengths. A common example of standing waves are the waves produced by stringed musical instruments. When the string is plucked, pulses travel along the string in opposite directions. The ends of the strings are fixed in place, so nodes appear at the ends of the strings—the boundary conditions of the system, regulating the resonant frequencies in the strings.

The resonance produced on a string instrument can be modeled in a physics lab using the apparatus shown in Figure. The string has a node on each end and a constant linear density. The length between the fixed boundary conditions is L. The hanging mass provides the tension in the string, and the speed of the waves on the string is proportional to the square root of the tension divided by the linear mass density.

The lab setup shows a string attached to a string vibrator, which oscillates the string with an adjustable frequency f. The other end of the string passes over a frictionless pulley and is tied to a hanging mass. The magnitude of the tension in the string is equal to the weight of the hanging mass. The symmetrical boundary conditions a node at each end dictate the possible frequencies that can excite standing waves.

The fundamental frequency , or first harmonic frequency, that drives this mode is. A node occurs at each end of the string. The nodes are boundary conditions that limit the possible frequencies that excite standing waves. Note that the amplitudes of the oscillations have been kept constant for visualization.

The standing wave patterns possible on the string are known as the normal modes. Conducting this experiment in the lab would result in a decrease in amplitude as the frequency increases.

The equations for the wavelength and the frequency can be summarized as:. The standing wave patterns that are possible for a string, the first four of which are shown in Figure , are known as the normal modes , with frequencies known as the normal frequencies. In summary, the first frequency to produce a normal mode is called the fundamental frequency or first harmonic.

Any frequencies above the fundamental frequency are overtones. The solutions shown as Equation and Equation are for a string with the boundary condition of a node on each end. When the boundary condition on either side is the same, the system is said to have symmetric boundary conditions.

Equation and Equation are good for any symmetric boundary conditions, that is, nodes at both ends or antinodes at both ends. The waves produced by the vibrator travel down the string and are reflected by the fixed boundary condition at the pulley.

There is a node on one end, but an antinode on the other. The three standing modes in this example were produced by maintaining the tension in the string and adjusting the driving frequency. Keeping the tension in the string constant results in a constant velocity.

The same modes could have been produced by keeping the frequency constant and adjusting the speed of the wave in the string by changing the hanging mass. Visit this simulation to play with a 1D or 2D system of coupled mass-spring oscillators. Vary the number of masses, set the initial conditions, and watch the system evolve.

See the spectrum of normal modes for arbitrary motion. See longitudinal or transverse modes in the 1D system. The equations for the wavelengths and the frequencies of the modes of a wave produced on a string:.

These modes resulted from two sinusoidal waves with identical characteristics except they were moving in opposite directions, confined to a region L with nodes required at both ends. Will the same equations work if there were symmetric boundary conditions with antinodes at each end? What would the normal modes look like for a medium that was free to oscillate on each end?

Yes, the equations would work equally well for symmetric boundary conditions of a medium free to oscillate on each end where there was an antinode on each end. The normal modes of the first three modes are shown below. The dotted line shows the equilibrium position of the medium.

Note that the first mode is two quarters, or one half, of a wavelength. The second mode is one quarter of a wavelength, followed by one half of a wavelength, followed by one quarter of a wavelength, or one full wavelength.

The third mode is one and a half wavelengths. These are the same result as the string with a node on each end. The equations for symmetrical boundary conditions work equally well for fixed boundary conditions and free boundary conditions. These results will be revisited in the next chapter when discussing sound wave in an open tube. The free boundary conditions shown in the last Check Your Understanding may seem hard to visualize.

How can there be a system that is free to oscillate on each end? In Figure are shown two possible configuration of a metallic rods shown in red attached to two supports shown in blue. In part a , the rod is supported at the ends, and there are fixed boundary conditions at both ends.

Given the proper frequency, the rod can be driven into resonance with a wavelength equal to length of the rod, with nodes at each end. In part b , the rod is supported at positions one quarter of the length from each end of the rod, and there are free boundary conditions at both ends. Given the proper frequency, this rod can also be driven into resonance with a wavelength equal to the length of the rod, but there are antinodes at each end. If you are having trouble visualizing the wavelength in this figure, remember that the wavelength may be measured between any two nearest identical points and consider Figure.

When driven at the proper frequency, the rod can resonate with a wavelength equal to the length of the rod with a node on each end. When driven at the proper frequency, the rod can resonate with a wavelength equal to the length of the rod with an antinode on each end.

On the wave on a string, this means the same height and slope. Note that the study of standing waves can become quite complex. The answer is no. In this configuration, there are additional conditions set beyond the boundary conditions. Since the rod is mounted at a point one quarter of the length from each side, a node must exist there, and this limits the possible modes of standing waves that can be created.

We leave it as an exercise for the reader to consider if other modes of standing waves are possible. It should be noted that when a system is driven at a frequency that does not cause the system to resonate, vibrations may still occur, but the amplitude of the vibrations will be much smaller than the amplitude at resonance.

A field of mechanical engineering uses the sound produced by the vibrating parts of complex mechanical systems to troubleshoot problems with the systems. This may cause the engine to fail prematurely. The engineers use microphones to record the sound produced by the engine, then use a technique called Fourier analysis to find frequencies of sound produced with large amplitudes and then look at the parts list of the automobile to find a part that would resonate at that frequency.

The solution may be as simple as changing the composition of the material used or changing the length of the part in question. There are other numerous examples of resonance in standing waves in the physical world. The air in a tube, such as found in a musical instrument like a flute, can be forced into resonance and produce a pleasant sound, as we discuss in Sound.

At other times, resonance can cause serious problems. A closer look at earthquakes provides evidence for conditions appropriate for resonance, standing waves, and constructive and destructive interference. A building may vibrate for several seconds with a driving frequency matching that of the natural frequency of vibration of the building—producing a resonance resulting in one building collapsing while neighboring buildings do not. Often, buildings of a certain height are devastated while other taller buildings remain intact.

The building height matches the condition for setting up a standing wave for that particular height. The span of the roof is also important. Often it is seen that gymnasiums, supermarkets, and churches suffer damage when individual homes suffer far less damage. The roofs with large surface areas supported only at the edges resonate at the frequencies of the earthquakes, causing them to collapse.

As the earthquake waves travel along the surface of Earth and reflect off denser rocks, constructive interference occurs at certain points. Often areas closer to the epicenter are not damaged, while areas farther away are damaged. The resultant in the animation below is shown in black. The resultant is merely the result of the two individual waves - the blue wave and the green wave - added together in accordance with the principle of superposition. The result of the interference of the two waves above is a new wave pattern known as a standing wave pattern.

Standing waves are produced whenever two waves of identical frequency interfere with one another while traveling opposite directions along the same medium. Standing wave patterns are characterized by certain fixed points along the medium which undergo no displacement.

These points of no displacement are called nodes nodes can be remembered as points of no d e s placement. The nodal positions are labeled by an N in the animation above. The nodes are always located at the same location along the medium, giving the entire pattern an appearance of standing still thus the name "standing waves".

A careful inspection of the above animation will reveal that the nodes are the result of the destructive interference of the two interfering waves.