Physics: Simple Harmonic Motion, Waves & Sound (AP Physics)

When you see waves in a pond, string of a guitar being strummed or sounds from waves crashing against the rocks, what you are seeing is wave formation. So, waves are a result of disturbance of equilibrium of a system that travel from one part of the system to the other. Here the medium could be anything – liquid, solid or gas. Often such waves are called mechanical waves

The wave function helps us establish the relationship between displacement of the particle of a medium in y direction with the position of the particle relative to the origin and time t. In a way, it helps us predict the behavior of a wave.

Transverse wave and wave equation form the basis of understanding the chapter on waves. Learn how the wave velocity and particle velocity of the medium differ. Learn also what is phase angle and phase of a wave.

Derivation of wave velocity in X direction can bring a lot of clarity to the concept of using the formula. In this 4th lesson on waves, what you will learn is one, what is the velocity of a wave as it moves forward and two, what is the velocity and acceleration of “a point” on a wave in a direction perpendicular to the motion of the wave, at different times.

Derivation of wave velocity equation can be done using several methods. The simplest is using Newton's second law of motion. We have established that the speed of a wave is related to the wave’s wavelength and frequency but if you think a little deeper, what you will find is that eventually it is set by the properties of the medium. So you’ll never find a thread between your hands vibrate the same way as a rubber band, if you’ve pulled both to the same magnitude of force and plucked them pretty much the same

Kinetic energy and potential energy of a wave is a function of mass of the string and the elasticity of the string. Consider part of a string and say its mass is dm. As the wave passes through it, the mass oscillates transversely in simple harmonic motion and there is kinetic energy associated with it due to the “transverse velocity” or the velocity in the y direction.

We must understand that to create a sinusoidal wave along a straight string, the wave needs to create a stretch in the string. And we know that a length change or stretching of the string automatically results in generation of elastic potential energy, quite like a spring.

Superposition principle and interference of waves is a simple concept that explains how waves interfere . You can find more simulations at University of Colorado site (https://phet.colorado.edu). 

The principle of superposition says that overlapping waves can be added algebraically to produce a resultant wave or a net wave. While, the combining of waves is what physicists call waves interference, the waves are said to interfere. You may note that these terms refer only to the “wave displacements”, the travel of waves is pretty much unaffected. The resultant wave is also a sinusoidal wave that moves in the same direction. Constructive interference occurs when the phase difference between the waves is an even multiple of π (180°), whereas destructive interference occurs when the difference is an odd multiple of π. If the difference between the phases is intermediate between these two extremes, then the magnitude of the displacement of the summed waves lies between the minimum and maximum values.

Learn in the numerical problem how 2 identical waves interfere

Standing waves and resonance along with the concept of nodes and anti nodes are often connected with sound also. If two sinusoidal waves having the same amplitude and wavelength travel in opposite directions along a stretched string, they can interfere to produce a standing wave. Unlike a traveling wave, a standing wave does not transfer energy from one end to the other. For “certain frequencies” the interference of waves produces a standing wave pattern or you could say an oscillation pattern with nodes and anti nodes like this. And When this happens, we say that a standing wave has been created and the chord or string is said to be resonating at these certain frequencies, that are called resonant frequencies. Well, physicists call this oscillation mode with the lowest frequency as the fundamental mode or the first harmonic when n=1. Then the second harmonic is the oscillation mode with n = 2, the third harmonic is with n= 3, and so on and so forth. Also, the associated frequencies are called f1, f2, f3, and so on and a collection of all possible oscillation modes is labeled the harmonic series, and n is called the harmonic number of the nth harmonic.

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Sound waves are a type of longitudinal wave. Longitudinal waves and transverse waves behave quite similarly mathematically.  However, at a visual level the vibration of particles of the medium differs in terms of weather it is parallel to the direction of the motion of the wave or perpendicular. In longitudinal waves, the motion of the particle is parallel to the direction of the wave. This is opposite of what happens in transverse wave where it is perpendicular to the wave.

Displacement and pressure in a sound wave describe such waves and compression and rarefaction are two other characteristics . The mathematical treatment of longitudinal waves is almost the same as transverse waves but with some minor changes in the way we look at such waves visually. What we will learn in this lesson are two things. One- what is the equation that defines a longitudinal wave and how do we interpret it, and two - why do we call such waves “pressure waves” and what is the equation that defines a pressure along the wave. So a longitudinal wave is a repeating pattern of compression and rarefaction and the wavelength is measured as the distance from one compression to the next adjacent compression or the distance from one rarefaction to the next adjacent rarefaction.

Interference of sound waves (longitudinal) happens much the same way as transverse waves. It depends on the phase difference between the two. Sound waves or longitudinal waves can undergo interference, quite the way transverse waves interfere. So let us say there are two sound waves that have same amplitude and wavelength, but have a phase difference of phi and they travel in the positive direction of x axis. Then the resultant wave could be due to fully constructive interference or a flat wave with zero amplitude due to destructive interference.

Sound intensity and the decibel scale indicate the mathematical measure of loudness of sound. It is indicated by symbol I and is the power transmitted per unit area by sound waves. It depends on the density of the medium, v is the velocity of the wave, omega is the angular frequency of the wave and Sm is the amplitude of vibrations. We can use logarithmic scale to describe intensity of sound as well so that instead of describing intensity I as large numbers going into trillions, we can make it a lot more convenient by handling smaller numbers. And these numbers are called sound levels or decibels

See in the numerical problem how drummers use electronic devices to reduce the intensity of sound around them

Well, the physics behind creation of sound waves in a pipe filled with air is much like the creation of standing waves in a string or a transverse wave. The only difference is that in transverse waves the particles of the medium move perpendicular to the direction of the motion of the wave, but in a pipe that has air, the motion of the particles of a standing wave is parallel to the direction of the wave.

So if we apply this theory to stringed instruments or pipe instruments, we can say that the length of a musical instrument determines the frequencies that can be produced in that instrument. So if you take an instrument like this one that’s a violin and has short strings or length L, you will find it has a sound that has high frequency and if you take one that has longer strings, say a cello, it will be low frequency output

So when you are in a plane, sometimes you experience a kind of humming that goes up and down so what do you think is happening? Well, it’s actually the two engines emitting sound at near same frequency resulting in a beat phenomenon.

Think of a time when you saw an ambulance or a police car approach you and then went past you blaring the siren. You would have observed the intensity of sound go up and then down. The increase and decrease in frequency that you as an observer experience would depend on the speed at which the police car moves towards you and then moves away from you. Well if you are also moving towards the police car or let’s say away from it, the frequency of sound you hear would depend on the “relative speed” between you and the car.

So, these changes in frequency on account of relative motion between a source which is the police car in this case and the observer that is you is what we call the of the Doppler effect. The effect was found by an Austrian physicist Johann Doppler in 1842 and infact Doppler effect is true not only for sound waves but also for other kind of waves like microwaves, radio waves or electromagnetic waves in general and for visible light as well. But, in this lesson we will discuss doppler effect for sound waves in air only