### Appendix 4.1: Elasticity of the Air

The air contained in the body is an elastic and compressible medium.

Consider a rigid-wall container (or *cavity*, as we called it) where the shape is irrelevant. The cavity encloses a volume of air **V**_{0}, initially at standard ambient pressure. Then suppose we can apply a force from outside into the cavity, that by compressing the air causes a rise in pressure and a consequent contraction of the air volume.

The situation is sketched below.

A parameter

**K** (also called

*bulk modulus*, which measures a substance resistance to uniform compression) tells us how the volume

*∆V* changes with respect to the initial volume

*V*_{0} under a pressure variation

*∆P* (with respect to standard ambient pressure):

$$\Delta p = - K\frac{\Delta V}{{V_{0} }}$$

The negative mark indicates that volume diminishes when pressure grows.

The bulk modulus **K** recalls the elastic modulus **E** (or Young’s modulus) whose physical definition was given in Chap. 3 regarding the plate oscillator. Both the **K** and the **E** modulus describe the elastic properties of a substance, with some important differences.

The elastic modulus **E** indicates how much a substance (like aluminium or wood) elongates when undergoing a pulling force **F**. Orthotropic materials (like wood) go through different deformation depending on the sense of pulling (along or across the grain); that is why we have defined two elastic moduli for orthotropic plates (*E*_{x} along the grain and *E*_{y} in the perpendicular sense).

On the other hand, the bulk modulus

**K** tells us how much an air volume diminishes when undergoing an increase in pressure

*∆P.* Since air is a homogeneous substance, the

**K** modulus does not depend on the direction of the force that modifies both pressure and volume; instead it is constant and corresponds to the product of the density of the air

**ρ** and the square of the sound speed

**c**, that is

From this formula we can obtain the value of the bulk modulus.

Being

*c* = 343.3 m/s and

*ρ* = 1.204 kg/m

^{3}, the result in standard MKS units is

$$K = 1.42 \times 10^{5}$$

The air contained in the rigid wall cavity of volume

**V**_{0} behaves like a spring under a compressing (or expanding) force. As formerly observed, the lengthening of a spring under a force

**F** applied at one end is

*x* =

*F/k* and

**K** is the stiffness of the spring; the higher the stiffness

**K** the minor the lengthening. The equivalent parameter applied to a cavity enclosing a volume of air is the mechanical compliance

**C**_{m} which is the reciprocal of the stiffness and is defined as

$$C_{m} = \frac{V}{{KS^{2} }} = \frac{V}{{c^{2} \rho S^{2} }}$$

where S is the surface on which the force is applied. So this is the parameter we need to evaluate the characteristics of the oscillation in the cavity.

While—as seen in Chap.

3—the natural frequency of a mass-spring system is

$$f = \frac{1}{2\pi }\sqrt {\frac{k}{m}}$$

the natural frequency of a cavity is

$$f = \frac{1}{2\pi }\sqrt {\frac{1}{{C_{m} m}}}$$

where

*m* in this case is the mass of the air volume, therefore

\(m = V \times \rho\).

### Appendix 4.2: Air Frequency in Mode 〈0 1〉

In mode 〈0 1〉 the pressure wave in a guitar-shaped cavity develops along the longitudinal axis and propagates at velocity

**c** of the sound in the air

\((c = 343.3\,{\text{m}}/{\text{s}})\)). The wave travels as a half sinusoid through the length

**l** of the body. In other words this is a

*longitudinal half*-

*wave mode of vibration*. So in this mode the wave traces a full sinusoid over a distance

*twice* the length

**l** of the body. This span (between two consecutive maximum points of the wave) is called

*wavelegth* **λ**, which in this vibration mode is written as

The wave inside the cavity covers the distance between two consecutive maximum or minimum points in a time

**T**, and the relation between the three units involved (propagation velocity c, covered distance

**λ** and time

**T**) is plainly

$$velocity = \frac{travel\,length}{time}\quad {\text{therefore}}\,\,\,\, c = \frac{\lambda }{T} = \frac{2l}{T}$$

But the time **T** required for the wave to trace a full sinusoid *at the natural frequency associated with this particular vibration mode* (mode 〈0 1〉) is also the period of this frequency, or *T* = *1*/*f.*

By combining these correlations we can determine the theoretical formula for the natural frequency of the cavity in mode 〈0 1〉:

$$f = \frac{c}{\lambda } = \frac{c}{2l}$$

The formula \(f = \frac{c}{\lambda }\) is valid for all of the vibration modes we are concerned with in this text, provided that, case by case, the wavelength **λ** is duly associated with one of the instrument dimensions. This connection will be reviewed in each of the modes we are going to deal with.

### Appendix 4.4: Helmholtz Resonance

The Helmholtz resonator behaves like an oscillating mass-spring system whose operation model is:

As we have observed, the natural frequency in this system is

$$f = \frac{1}{2\pi }\sqrt {\frac{k}{m}}$$

where

**k** is the stiffness of the spring,

**m** is the mass and

**r** is the parameter that sums up the energy losses in the system.

In the Helmholtz resonator

Mass corresponds to the weight of the air in the neck, so

*Mass* = *length of the neck* × *surface* × *density of the air* \((m = l\,S\,\rho )\)

The elastic component is derived from the volume of the air in the body. In Appendix 4.1 we provided the mathematical expression of the compliance of the air volume (which is the reciprocal of the stiffness):

$$C_{m} = \frac{V}{{KS^{2} }} = \frac{V}{{c^{2} \rho S^{2} }}$$

where S is the surface of the opening. Introducing the expressions of mass and stiffness into the frequency general formula of a mass-spring system we get the traditional expression of the Helmholtz resonator natural frequency:

$$f_{H} = \frac{c}{2\pi }\sqrt {\frac{S}{lV}}$$

For thin-walled resonators or for those where the openings have a particular shape (like the *ff* in the violin) the parameter *l* must be replaced by the parameter *l*_{effective} which considers how the air flows into the opening forming a sort of virtual channel.

In the case of a circular opening the effective length of the resonator neck was calculated as follows:

(Rayleigh—

**Theory of Sound**—1878):

$$l_{actual} = l_{0} + \frac{\pi }{2}r$$