Waves
There are many different kinds of waves -- water waves, waves on a string, sound waves and light waves -- that all share common features:
- waves transmit energy without transmitting mass
--> waves are a moving "disturbance" through a "medium"
- wave disurbances result in vibrations of the "elements" of a medium that are communicated (through Newton's 2nd Law) to neighboring elements
- traveling waves can be characterized by
- a frequency (f) of oscillation of the elements of the medium. Graphically, at a fixed position along the medium, a part of the medium moves in time as shown:
- a wavelength, or a characteristic distance scale along the medium where the disturbance pattern repeats itself. Graphically, at a fixed instant in time (imagine using a camera to take a snapshot of the medium, freezing the time), the medium looks like
Waves travel through a medium with a speed characteristic of the medium. Let's see what this means by considering sound waves traveling through air. Different pure tone sounds have different frequencies, leading to a perceived quality of sound known as pitch. A high-pitched sound has a high frequency: for example, the highest key on a piano is a C note, 4 octaves higher than middle C, producing a sound with a frequency of 4186 Hz. A low-pitched sound has a low frequency: the lowest C note on the piano is 3 octaves below middle C and produces a sound with a frequency of 32.7 Hz. When middle C is played on the piano, it produces a sound with a frequency of 262 Hz. Clearly, these sounds have very different frequencies. Nonetheless, they all travel through air at the same speed, vsound=344 m/s (for air at the temperature 21oC). Otherwise, it would be a very different experience watching and listening to someone play the piano!
In general, for all waves, the wave speed is a characteristic of the medium: water waves move with a speed characteristic of the water, waves on a string move with a speed depending on the tension and mass of the string (more on this below), and sound waves move through air at 344 m/s. In addition, the wave speed is related to the frequency and wavelength (
) in the following way:
What this means is that the wavelength and frequency of a wave are not independent. Their product yields a constant, characteristic of the medium but not depending on the wave disturbance. Hence, high frequency waves must have short wavelengths and low frequency waves must have long wavelengths. An example might help:
Let's consider a wave pulse on a string. The way we'll make the pulse is to stretch a string between two fixed points, and then jolt one end. We'll consider two different sections (1 and 2) of the string as shown in the picture below: one that includes the pulse and one that is just beyond the pulse.
Each section pulls on the other with equal magnitude, but oppositely directed forces. This results from another of Newton's Laws (Newton's 3rd law). Section 2 of the string is pulled up by section 1 while section 1 is pulled down by section 2, meaning that a little later in time, the pulse has moved to the left by a little bit. The speed at which the disturbance moves depends on the force exerted by one section on the other (e.g., the tension in the string) and the section's resistance to acceleration (e.g., it's mass). From this, we can deduce that the wave speed is:
What this means for waves on a string is that we can increase the wave speed by pulling harder on the string. Increased tension leads to higher wave speed. We'll see in a few days that this is how we tune a guitar string to produce sound of a desired frequency.
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Last updated:
29 Sep 1999
Comments: bland@indiana.edu