Sound Wave Length
and Propagation
The length of a sound wave is determined by its frequency and the speed of sound propagation. While the speed of sound propagation varies with different air temperatures, for calculations, room temperature and a sound speed of 343 m/s can be used. The length of a sound wave (λ) can be calculated using the following formula: λ = v / f. Where: λ - sound wave length (in meters), v - speed of sound in the medium (meters per second), f - sound frequency (hertz).
For example, at a frequency of 440 Hz (A4 note frequency), λ = 345 m/s / 440 Hz ≈ 0.78 meters. Sound wave length based on frequency:
Frequency (Hz) Sound Wave Length (meters)
- 20 17.15
- 40 8.58
- 80 4.29
- 160 2.14
- 320 1.07
- 640 0.53
- 1280 0.27
- 2560 0.13
- 5120 0.07
- 10240 0.03
- 20480 0.01
In everyday use, it's easier to work with millimeters and milliseconds. Sound travels 345 mm in one millisecond (1/1000 of a second). Therefore, for a frequency of 1000 Hz, the wavelength is 345 mm, halving for every doubling of frequency.
Five important phenomena are related to wave length. The first is resonance, where a sound wave doesn't dissipate because it matches the system's own oscillation frequency. The resonant frequency depends on mass and stiffness for solid objects, while in a room, the ratio of room dimensions to wavelength is crucial. Everything has a vibration frequency – houses, floors, windows, and even a loudspeaker's cabinet or diaphragm.
For solid objects, the resonant frequency in a closed space is any dimension's half wavelength and its multiples. This is why bass from a loudspeaker may sound fine in one room and problematic in another. Resonance is a challenging phenomenon in music reproduction, as a loudspeaker's response in a real room looks like a roller coaster, especially due to resonance. Resonance in a loudspeaker's cabinet, sides, and diaphragm can also cause disturbances in the sound, both directly and harmonically. It's challenging to dampen resonances in rooms. Helmholtz resonators can be used; for example, a suspended ceiling, wardrobe door, or a similar solution.
The second phenomenon is sound wave reflection. When a sound wave encounters an object, sound reflection can occur. In the case of sound reflection, the sound wave bounces off the object at an angle equal to the angle of incidence (similar to how light reflects off a mirror). Reflections of sound waves are one source of room and loudspeaker cabinet resonances, leading to standing waves.
The reflection of sound waves from an object is influenced by the object's size relative to the wavelength. Sound waves reflect most efficiently when the object's size is comparable to or larger than the sound wave's wavelength and fade smoothly when the object is about 10 times smaller than the wavelength. Fortunately, human hearing can distinguish between the direct sound that arrives first, the reflected sound, and the delayed sound, treating the latter as secondary and unimportant. Still, reflections can significantly alter the sound image. When the distance traveled by sound directly and via reflection differs by half, one and a half, two wavelengths, etc., silence occurs in that spot because the sounds cancel each other out. Similarly, if the distance differs by one wavelength or its multiple, amplified sound occurs, as the sound waves add up in-phase. Generally, reflections are an issue when the reflection point is close to and directly in line with the loudspeaker, and the strengths and timing of the reflected and direct sounds vary only slightly. Floor reflections are often the biggest problem, as moving the loudspeaker away from the floor is challenging, and it is usually closer than the ceiling.
Furniture and soft furnishings significantly reduce room reflections and scatter them. Furniture can also help absorb and diffuse sound reflections, just as concert hall walls use angled panels of different sizes to achieve diffusion. While sound may still reflect, it does so unevenly, leading to dispersion.
The same mutual damping and reinforcement of sound waves occurs when a loudspeaker's drivers are at an angle to the listener. For instance, when moving from a seated to a standing position, the upper loudspeaker moves closer to the listener, and the lower one moves farther away. At the transition frequency of the upper driver, waves interfere, amplifying some frequencies and cancelling others. To reduce this phenomenon, higher-frequency and shorter-wavelength drivers are placed close together. For bass, this is less crucial due to the much greater wavelength compared to the size of the space.
As all sounds eventually decay, we must discuss sound absorption. The conservation of energy applies to sound waves, and the only way to eliminate sound waves is to convert them into another form of energy (heat). Sound absorption should be distinguished from reducing sound transmission; even a perfectly sound-reflecting object might let a little sound pass through without absorbing it. For sound absorption, three conditions must be met. Firstly, the material should vibrate as much as possible with the sound wave. Secondly, there should be a significant amount of actual mass vibration. Thirdly, the material's thickness relative to the sound wave length should be noticeable, often considered to be at least 1/4 to 1/3 of the wavelength. For example, loudspeakers use fine fibrous mass to dampen sound waves, often polyester. However, it's hard to imagine anyone wanting to attach half a meter thick fleece to their wall just to eliminate a 150 Hz reflection. Nevertheless, sound reflections need to be considered, and sometimes the best investment in sound quality is new furniture or a rearranged room.
Another important topic is sound shielding and diffraction. Shielding happens when a sound wave's wavelength is smaller than the size of an object, causing most of the sound to reflect or be absorbed, and not reaching the area behind the object (similar to a shadow in light). Diffraction is the bending of a sound wave at an object's edge or corner. The first place we encounter diffraction is in the loudspeaker itself. As sound is supposed to propagate spatially in all directions, the loudspeaker's front surface is where sound cannot propagate backwards until it reaches the loudspeaker's edge, where the sound wave bends, and the edge becomes a new sound source. Due to the certain distance to the edge, the sound generated by this new source may be out of phase, amplifying or canceling the direct sound from the loudspeaker diaphragm.
To mitigate this phenomenon, the best loudspeakers do not have uniformly shaped enclosures but have, for example, uneven width, substantially rounded edges, separate placement of high-frequency drivers, or other ways to acoustically address the issue. Of course, the distribution filter in the loudspeaker can also correct this phenomenon, but it is always better to eliminate the problem rather than correct it. The same phenomenon occurs with other objects that sound hits and that have sharp edges relative to the sound wave's wavelength.
If an object is significantly smaller than the wavelength, it becomes acoustically negligible. Therefore, for a 20 Hz sound with a 17-meter wavelength, all objects in the room are acoustically negligible and do not hinder sound propagation. This is why manufacturers allow placing subwoofers under beds, tables, or other inconspicuous locations. Similarly, the loudspeaker cabinet itself becomes invisible to low frequencies, which propagate just as freely behind the cabinet as in front of it. As the wavelength of a frequency range becomes equal to the loudspeaker's diameter, the sound starts radiating more directly. To compensate for this transition, the front plate of the high-frequency driver is often designed as a waveguide to ensure a smooth transition. A waveguide always has the role of smoothing out transitions, but it also tries to spread even the highest frequencies slightly to widen and even out the sound image.
The scattering of waves due to reflections was briefly discussed and is especially important in room acoustics.
There are several observations for loudspeaker buyers:
To identify panel resonances of a loudspeaker, listen to a song with your ear pressed airtightly against the loudspeaker panel. If you hear a subdued, uniform sound in the lower range of frequencies, the resonances of the loudspeaker panels are likely under control. Absolute silence can only be heard from loudspeakers made of exceptionally rare and heavy materials.
To reduce internal resonances, the loudspeaker cabinet is made with varying dimensions in all directions, so that no dimension is an integer multiple of another. Higher-end loudspeakers have sides at angles to each other, causing reflected sound waves to dissipate as they circulate within the cabinet and not influence the audible sound from the diaphragm. This is particularly important for bass and midrange frequencies. The high-frequency driver is typically placed in its own isolated enclosure, and if the manufacturer is reputable, resonances in the high-frequency range are controlled as well.
Detecting problems related to loudspeaker diffraction is easier. If the loudspeaker is a large, boxy enclosure, the problem is present. Slight asymmetry and rounding suggest that efforts have been made to address the issue. The easiest way to detect this is by listening to the loudspeaker from different angles. High frequencies should decrease at angles, but uniformly. This way, the loudspeaker doesn't become a problem in real listening conditions, and it can be enjoyed from various positions. A good loudspeaker remains audible even at wide angles, just with reduced sharpness and detail.
These observations can lead to specific guidelines:
Choose a loudspeaker with an unconventional shape, as a square box indicates cost-cutting, and achieving high quality from such an enclosure is mostly down to luck.
Ensure that the high-frequency driver's front plate is uniquely shaped and the distance to the edges is uneven or that the high-frequency driver is housed separately. Make sure the high-frequency and mid-frequency drivers are placed close together.
