A hi-fi tweeter has a rectangular aperture 5 by \(12 \mathrm{~cm}\). Which way would you mount this so that the sound pattern is broad in the horizontal plane and narrow in the vertical? What is the approximate radiation pattern at \(10,000 \mathrm{~Hz}\) ?

Sound wave diffraction involves the bending of waves around obstacles and the spreading out of waves through small openings. This principle explains how sound can be heard even if the source is obscured by an object or how it emerges from a small speaker aperture. The extent of diffraction depends on the relationship between the wavelength of the sound and the size of the opening or obstacle. Typically, when the wavelength is comparable to or larger than the aperture, diffraction is more significant, leading to a wider radiation pattern. This concept plays a significant role in designing speaker systems, as it will influence how the sound is distributed in a room.

The frequency and wavelength of a wave are inversely proportional, where the wavelength is the distance between two consecutive peaks of a wave and the frequency is how often these peaks pass a point per second. This relationship is critical in sound because it determines how sound waves propagate through a medium. Given a constant speed of sound, a higher frequency sound (like a tweeter producing a high-pitched note) results in a shorter wavelength. When configuring audio equipment, understanding this relationship helps in predicting how the sound will interact with the environment, especially when considering diffraction and the radiation pattern of a speaker.

The speed of sound is the rate at which sound waves travel through a medium. In standard air conditions at sea level, this is approximately 343 meters per second. However, the speed can vary based on factors like temperature, humidity, and air pressure. This speed is crucial for calculating the wavelength of a sound wave when the frequency is known, using the formula \( \lambda = \frac{c}{f} \). This parameter also influences the arrival time of sound at different distances and hence is an important consideration when synchronizing audio playback in speaker arrays or theatre systems.

Mounting orientation for speakers can have a significant impact on how the sound is distributed in a given space. The orientation dictates the direction of the broader and narrower radiation patterns, which should be strategically positioned according to the listening environment. For example, a speaker like a tweeter, which produces high-frequency sounds with short wavelengths, can be mounted with its longest dimension along the horizontal plane to create a wider soundscape, which is often desirable for music and movie experiences in home theatre systems or sound stages.

The radiation angle calculation is used to determine the spread of sound from a speaker's aperture. It considers the wavelength of the sound and the size of the speaker opening. The formula to estimate the radiation angle is \( \theta \approx \frac{\lambda}{D} \), where \( \theta \) is the radiation angle, \( \lambda \) is the wavelength, and \( D \) is the dimension of the opening. This calculation helps audio engineers and enthusiasts tailor the sound distribution, ensuring optimal audio quality and coverage for the intended listening environment. When expressed in radians, as in the given solution, the angle can be converted to degrees for a more intuitive understanding by multiplying by \( \frac{180}{\pi} \).