How can standing waves be minimized

The waves move through each other with their disturbances adding as they go by. If the two waves have the same amplitude and wavelength, then they alternate between constructive and destructive interference. The resultant looks like a wave standing in place and, thus, is called a standing wave. Consider two identical waves that move in opposite directions. The waves interfere and form a resultant wave. Notice that the resultant wave is a sine wave that is a function only of position, multiplied by a cosine function that is a function only of time.

Graphs of y x , t as a function of x for various times are shown in Figure The red wave moves in the negative x -direction, the blue wave moves in the positive x -direction, and the black wave is the sum of the two waves.

As the red and blue waves move through each other, they move in and out of constructive interference and destructive interference. In fact, the waves are in phase at any integer multiple of half of a period:. This happens at. Notice that some x -positions of the resultant wave are always zero no matter what the phase relationship is. These positions are called node s.

Where do the nodes occur? Consider the solution to the sum of the two waves. Finding the positions where the sine function equals zero provides the positions of the nodes. These are the antinode s. What results is a standing wave as shown in Figure The resulting wave appears to be a sine wave with nodes at integer multiples of half wavelengths.

The resultant wave appears to be standing still, with no apparent movement in the x -direction, although it is composed of one wave function moving in the positive, whereas the second wave is moving in the negative x -direction.

The nodes are marked with red dots while the antinodes are marked with blue dots. 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 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. All frequencies above the frequency f 1 f 1 are known as the overtone s.

The equations for the wavelength and the frequency can be summarized as:. Figure 1 shows a low frequency measurement in a typical listening room. On the amplitude scale the Y axis, labelled "level" , we can see a very irregular response. The time scale the Z axis, labelled "time" is also very important. We can see that the resonances last a certain time.

Some call this ringing. Other rooms could easily exhibit longer ringing time. Have you ever sung in a bathroom? Some notes seem to make the whole room resonate.

In fact, this is exactly what happens. Standing wave is a low frequency resonance that takes place between two opposite walls as the reflected wave interferes constructively with the incident wave. The resonant frequency depends on the distance between the two walls. There will be many room modes between two walls as the phenomenon will repeat itself at multiples of the first frequency: 2 f , 3 f etc.

Between those two walls, there will be room modes at:. Fig 3 Distribution of the standing waves sound pressure level between two walls. Figure 3 illustrates the distribution of the standing waves sound pressure level between two walls. One can note that:. This explains why the low frequency response changes as we change the listening spot in a room.

It also explains why we have so much bass close to a wall. The phenomenon is unavoidable; there are standing waves even in the best rooms.

Standing Waves Sometimes waves do not seem to move; rather, they just vibrate in place. The vibrations from the fan causes the surface of the milk of oscillate. The waves are visible due to the reflection of light from a lamp. The resulting wave is shown in black.

Consider the resultant wave at the points and notice that the resultant wave always equals zero at these points, no matter what the time is. Nodes appear at integer multiples of half wavelengths. Antinodes appear at odd multiples of quarter wavelengths, where they oscillate between The nodes are marked with red dots and the antinodes are marked with blue dots.

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.

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. Example Standing Waves on a String Consider a string of attached to an adjustable-frequency string vibrator as shown in Figure.

The string, which has a linear mass density of is passed over a frictionless pulley of a negligible mass, and the tension is provided by a 2.

Check Your Understanding The equations for the wavelengths and the frequencies of the modes of a wave produced on a string:. 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. Summary A standing wave is the superposition of two waves which produces a wave that varies in amplitude but does not propagate.

Nodes are points of no motion in standing waves. An antinode is the location of maximum amplitude of a standing wave. Normal modes of a wave on a string are the possible standing wave patterns. The lowest frequency that will produce a standing wave is known as the fundamental frequency.

The higher frequencies which produce standing waves are called overtones. Key Equations Wave speed Linear mass density Speed of a wave or pulse on a string under tension Speed of a compression wave in a fluid Resultant wave from superposition of two sinusoidal waves that are identical except for a phase shift Wave number Wave speed A periodic wave Phase of a wave The linear wave equation Power in a wave for one wavelength Intensity Intensity for a spherical wave Equation of a standing wave Wavelength for symmetric boundary conditions Frequency for symmetric boundary conditions.

Conceptual Questions A truck manufacturer finds that a strut in the engine is failing prematurely. Consider a standing wave modeled as Is there a node or an antinode at What about a standing wave modeled as Is there a node or an antinode at the position? A 2-m long string is stretched between two supports with a tension that produces a wave speed equal to What are the wavelength and frequency of the first three modes that resonate on the string?

The length of the string between the string vibrator and the pulley is The linear density of the string is The string vibrator can oscillate at any frequency. A cable with a linear density of is hung from telephone poles. The air temperature is What are the frequency and wavelength of the hum? Consider two wave functions and.

Write a wave function for the resulting standing wave. The linear mass density of the string is and the tension in the string is The time interval between instances of total destructive interference is What is the wavelength of the waves? The resonance mode of the string is produced.

Write an equation for the resulting standing wave. Additional Problems Ultrasound equipment used in the medical profession uses sound waves of a frequency above the range of human hearing. If the frequency of the sound produced by the ultrasound machine is what is the wavelength of the ultrasound in bone, if the speed of sound in bone is. Shown below is the plot of a wave function that models a wave at time and. The dotted line is the wave function at time and the solid line is the function at time.

The speed of light in air is approximately and the speed of light in glass is. A red laser with a wavelength of shines light incident of the glass, and some of the red light is transmitted to the glass. The speed of sound of sound in air is if the air is at a temperature of.

What is the wavelength of the sound? A motorboat is traveling across a lake at a speed of The boat bounces up and down every 0. Use the linear wave equation to show that the wave speed of a wave modeled with the wave function is What are the wavelength and the speed of the wave? Given the wave functions and with , show that is a solution to the linear wave equation with a wave velocity of. A transverse wave on a string is modeled with the wave function. A sinusoidal wave travels down a taut, horizontal string with a linear mass density of The magnitude of maximum vertical acceleration of the wave is and the amplitude of the wave is 0.

The string is under a tension of. The wave moves in the negative x -direction. Write an equation to model the wave. A transverse wave on a string is described with the equation What is the tension under which the string is held taut? A transverse wave on a horizontal string is described with the equation The string is under a tension of The range finder was calibrated for use at room temperature , but the temperature in the room is actually Assuming that the timing mechanism is perfect, what percentage of error can the student expect due to the calibration?

Emergency stop. A traveling wave on a string is modeled by the wave equation The string is under a tension of Consider two periodic wave functions, and a For what values of will the wave that results from a superposition of the wave functions have an amplitude of 2 A? Consider two periodic wave functions, and. The height of the water waves are modeled with two sinusoidal wave equations, and What is the wave function of the resulting wave after the waves reach one another and before they reach the end of the trough i.

Hint: Use the trig identities and. If they traveled the same path at constant wave speeds of and how far away is the epicenter of the earthquake? A string with a linear mass density of is attached to the Consider the superposition of three wave functions and What is the height of the resulting wave at position at time.

One end of the string is fixed to a lab stand and the other is attached to a spring with a spring constant of The free end of the spring is attached to another lab pole.

The string is fixed at and Nodes appear at 2. The string has a linear mass density of Two resonant frequencies of the string are Hz and Hz. Challenge Problems A copper wire has a radius of and a length of 5. A pulse moving along the x axis can be modeled as the wave function a What are the direction and propagation speed of the pulse? Moves in the negative x direction at a propagation speed of.

A string with a linear mass density of is fixed at both ends. What is the wave function resulting from the interference of the two wave? Hint: and. The wave function that models a standing wave is given as. Plot the two wave functions and the sum of the sum of the two wave functions at to verify your answer.

The resultant wave form when you add the two functions is Consider the case where , and. Glossary antinode location of maximum amplitude in standing waves fundamental frequency lowest frequency that will produce a standing wave node point where the string does not move; more generally, nodes are where the wave disturbance is zero in a standing wave normal mode possible standing wave pattern for a standing wave on a string overtone frequency that produces standing waves and is higher than the fundamental frequency standing wave wave that can bounce back and forth through a particular region, effectively becoming stationary.

Previous: Next: 16 Chapter Review. Share This Book Share on Twitter. Resultant wave from superposition of two sinusoidal waves that are identical except for a phase shift. 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.