12b: Impedance

We already know that a wave which moves from one medium to another can change speed. In many cases this causes a bending of the wave called refraction. A second effect also occurs when a wave goes from one medium to another. There is usually a partial reflection of the wave, depending on how different the two mediums are. The resistance to the movement of a wave in a medium is called impedance and occurs for waves on a string, sound waves in air and electronic signals in a circuit. When a wave (of any kind) tries to travel from a medium with one impedance to a region where the impedance is different, there will be a partial reflection. The reflection means that not all the energy of the wave is transmitted to the new medium.

First look at the following simulation which shows what happens when a wave hits a solid boundary. Reflection can be of two types, depending on whether the boundary is ‘soft’ or ‘rigid’. If the wave is going from a more rigid medium into a softer medium the reflected wave will have the same phase as the incoming wave. Waves in a soft medium reflecting from a boundary with a stiff medium will change phase by 180 degrees.

Simulation exercise 12B (turn in answers on a separate sheet of paper): Reflection from Boundaries.

Now look at the following simulation which shows what happens when a wave passes from one medium into another. Some energy is reflected and the phase depends on whether the wave is traveling from soft to stiff or vice-verse. This example is for waves on a string but it also happens when sound waves inside an instrument reach the end of the instrument; some energy is reflected back into the instrument and some is transmitted to the outside.

Simulation exercise 12C (turn in answers on a separate sheet of paper): Impedance.

In some cases the goal is to avoid or minimize the effects of impedance. The process for doing this is to match impedance between the two regions. For example, the speakers of a stereo system are chosen to match the impedance of the amplifier/tuner component. In electrical components impedance is measured in Ohms (Ω) so if a speaker has an impedance of 10 Ω an amplifier with 10 Ω is chosen (or vice versa). If the impedance was not matched some of the energy from the amplifier would reflect back and not get to the speaker, making the system less efficient at turning electrical signals into sound.

Tube instruments have an impedance problem in that sound inside the tube will reflect off the ends, even an open end. Air inside the tube is confined by the walls of the tube but air outside is not. This difference in impedance causes some sound to reflect back into the tube at an open end and it is this reflected wave that sets up a standing wave inside the tube (remember from Chapter 7 that standing waves are made from two identical waves traveling in opposite directions). Reflection is strongest if the tube diameter is less than a quarter of the wavelength of the sound wave in the tube. Open end reflection is weaker as the diameter gets bigger compared to the wavelength.

On the one hand this reflection is what causes the tube to have a standing wave which gives rise to a fundamental frequency (resonance) but this also means less sound gets out of the instrument. Flutes are not corrected for impedance miss-match and are the softest woodwind instrument. Most other woodwinds are louder because they have bells that act as impedance matching devices between inside the instrument and outside. Brass instruments have even larger bells and so have better impedance matching and are even louder than woodwinds on average. As we will see, the bell of a wind instrument also affects the overtones present (and therefor the timbre) but the main function of a bell is to help sound waves exit the instrument by matching the impedance.

The mathematical symbol for impedance is Z, measured in ohms. In some references you will see the inverse of this number, the admittance which is defined as Y = 1/Z, measured in Siemens. Here is a more complete definition of acoustical impedance. (Note that the word impedance is used in many different contexts; electrical engineering in particular relates impedance to electrical signal transmission.)

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