Baffle step

For wavelengths below the width of the speaker diaphragm the acoustic energy from that speaker radiates in a spherical pattern. When the speaker is mounted on a baffle the radiation pattern becomes lobed towards the front. This is commonly observed by standing behind a speaker where there is noticeable drop in higher frequencies but no change in the bass response. With higher frequencies effectively being radiated into only the front hemisphere there is a perceived 6db drop in bass response, this is called baffle step. The step look like an S curve that begins to rise where the wavelength is 4 times the shortest distance to the edge of the baffle from the center of the speaker and reaches the top of the slope when the wavelength is 1/4 of the shortest distance to the baffle edge from the center of the speaker. The graph below represents the amplitude response for a speaker mounted on a sphere

Baffle Step Compensation

Compensating for the simple baffle step is achieved by applying a RC filter that mirrors the baffle step response and results in a flat response from the speaker. Unfortunately the baffle step is only smooth when the speaker is mounted on a sphere, in almost all other cases the step response is interfered by out of phase waves diffracted from the edge of the baffles.

Baffle Edge Diffraction

At frequencies below the baffle step the pressure of the sound wave simply bends around the baffle as if it wasn’t there. At frequencies above the baffle step the sound radiates uniformly till it reaches the edge the baffle where suddenly the acoustic impedance drops causing the sound energy to be divided and re-radiated from that point. According to Olson(1) this is similar to an incident ray of light diffracting into a reflected and transmitted wave when the medium that its traveling through changes density, (visual impedance?). The reflected wave has traveled across the baffle before it reaches you and so is delayed compared the sound that traveled straight from the speaker, this delay causes the two sounds to be out of phase and cancel each other out whenever the baffle width is a harmonic of the wavelength being played. This effect is most pronounced when the distance from the speaker to edge of the baffle is constant as all angles, as when the speaker is mounted on a circular disc. The graph below indicated the worst case scenario of baffle diffraction, the top wave is a phase graph and the bottom is amplitude response for speaker mounted on a 1meter wide disc.

This graph is for a speaker mounted on a rectangular baffle, the baffle step now the biggest influence but the irregularities in the higher frequencies are a result of the diffracted waves phase cancellation.

The effects of diffraction make the simple baffle step compensation ineffective because the response curve is no longer a simple S curve.

Solutions

One solution is to have the speaker mounted on a sphere; the change in acoustic impedance is gradual as the waves travel towards the “acoustic edge” of the sphere. The shape spreads out the diffraction uniformly resulting in a smooth S curve. Unfortunately building a sonically rigid sphere poses plenty of manufacturing challenges which result in high costs, some companies do offer this product and it almost solves the problem. A speaker can only play in a narrow frequency range requiring additional drivers to extend the frequency range, the distance between these drivers causes their sound waves to reach you at different times again resulting in phase issues. If one driver is mounted on a sphere then it is difficult to place an addition driver close enough to prevent phase issues. For a crossover point around 3 kHz a wavelength is about 10cm and if the centers of the two drivers playing that sound are further apart than that there will be interference. A solution needs to be on a flat surface to allow the mid-range and tweeter to be mounted as close as possible.

Spira Mirabilis

A speaker with a baffle whose edge has distance that is different at every angle will spread out the diffraction in a similar way that the sphere does, but being flat will also allow a tweeter to me mounted nearby. A spiral whose distance is a function a natural logarithm is called a Spira Mirabilis, miracle spiral, because at any point in the curve the angle to the next point in the curve is the same. The logarithmic spiral used in the Spira Mirabilis Signature speaker has a baffle edge distance that spans the wave lengths of the midrange drivers’ bandpass range. The tweeter is mounted less than the wavelength of the crossover point away from the midrange which created a very even dispersion with no combing effect and a large in-phase frontal lobe, a large “sweet spot”. Since the diffraction effects only occur at the higher frequencies the bass unit is a classic golden proportion enclosure with a tuned Helmholtz resonator.