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lecture3 - Lecture 3 Notes 06 29 Intro to waves A wave is...

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Lecture 3 Notes: 06 / 29 Intro to waves A wave is some kind of periodic disturbance that can move through a medium (waves on a string, sound waves in air, water waves in the ocean) or just through space (electromagnetic waves, matter waves in quantum mechanics). Waves in a medium propagate by particles pulling and pushing on each other. These waves can carry energy and momentum over long distances, as each particle transmits energy to the next, but individual particles move only slightly from the original position. Waves in a medium can be transverse (a string oscillating up and down, or an ocean wave on the surface of the water) or longitudinal (compression of a spring, or a sound wave in air): Waves can have all kinds of different shapes, but a particularly simple form is Note that the quantity k here has nothing to do with the spring constant (there aren't enough letters in all the alphabets of the world for physicists.) We will call these simple waves sine waves (whether they are a sine or a cosine). Note that a cosine is just a sine shifted over by an angle of π /2, so these two solutions are really the same thing. The plus or minus sign corresponds to left- or right-traveling waves, as we will see below. Here, y(x,t) is the value of the disturbance at position x and time t . In the case of a wave on a string oscillating up and down, it can be the height of a particular point on the string. In the case of a sound wave, it can be the change in the pressure of air at a particular place. The quantity θ = kx± ϖ t is called the phase of the wave. The quantity A is the maximum displacement attained by the wave, and is known as the amplitude .
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Motion of the wave First, let's see what the wave looks like at a fixed time t . This is a snapshot of a wave at some particular time: The horizontal axis is the distance (along a string, for example) and the vertical axis is the displacement of the wave at each point. The height of the wave crests is the amplitude A . The peak-to-peak distance is called the wavelength , and is denoted by the Greek letter λ . When one moves a distance , the wave goes through a full cycle. This means that the phase θ changes by 2 π . Since we are holding the time fixed, this means that The quantity k (again, not to be confused with the spring constant) is known as the wave number . It is equal to the number of radians by which the phase changes per meter, so its units are m -1 . Now, let's look at a particular point on the string, at position x, and follow its motion as a function of time. The graph of the displacement y at this point as a function of time looks something like this: The picture looks much the same as the snapshot of the wave at a fixed time, but now the horizontal axis is time, not position. The point moves with simple harmonic motion. The amount of time that passes between one instance of the displacement reaching its
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This note was uploaded on 09/19/2011 for the course PHYS 1C taught by Professor Smith during the Spring '07 term at UCSD.

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lecture3 - Lecture 3 Notes 06 29 Intro to waves A wave is...

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