Natural frequencies

Resonances, vibrations, together with natural frequencies, occur everywhere in nature. For example, one associates natural frequencies with musical instruments, with response to dynamic loading of flexible structures, and with spectral lines present in the light originating in a distant part of the Universe.

The simplest case of natural frequencies is illustrated by the vibration of a string. Its deflection $u ( x , t )$ satisfies boundary conditions, $u ( 0 , t ) = u ( \pi , t ) = 0$, and an initial condition, $u ( x , 0 ) = u_ 0 ( x )$. Its motion is described by the equation $\tau u _ { xx } = \rho u _ { t t }$. Separation of variables $u ( x , t ) = v ( x ) w ( t )$ leads to a pair of equations $v _ { x x } = \lambda v$, $w _ { t t } = \lambda w$.

In equations of the type $A \phi = \lambda \phi$, where $A$ is an operator whose domain is a certain class of functions, the number $\lambda$ is called an eigenvalue (cf. Eigen value), and $\phi$ is the corresponding eigenfunction. A (possibly complex) number $\mu$ is said to belong to the spectrum $\sigma ( A )$ of $A$ (cf. also Spectrum of an operator) if the "resolvent" operator $( A - \mu I ) ^ { - 1 }$ does not exist (cf. also Resolvent). $\mu = \lambda$ is an eigenvalue if it is a pole of $( A - \mu I ) ^ { - 1 }$, where $I$ denotes the identity operator. In the equation of a vibrating string, the boundary conditions are satisfied only if $\lambda$ is the square of a natural number $n$, $\lambda _ { n } = n ^ { 2 }$, $n = 1,2 , \dots$. The natural frequencies $\omega _ { n }$ are square roots of the eigenvalues: $\omega _ { n } = n$. The corresponding natural modes $\phi ( t )$ are the trigonometric functions $\operatorname { cos } nt$, $\operatorname { sin } nt$. The Euler formula $\operatorname { exp } ( i \alpha ) = \operatorname { cos } \alpha + i \operatorname { sin } \alpha$, with $i ^ { 2 } = - 1$, simplifies many arguments and offers a better insight into vibration and resonance, among others. The eigenvalues are real because in this case the operator $A$ is self-adjoint, meaning that for any pair $f , g$ in the domain of $A$, the inner product has the following symmetry: $\langle A f , g \rangle = \langle f , A g \rangle$ (cf. also Self-adjoint operator). In simple cases these products can be written as integrals.

As an example of this abstract theory, consider a free vibration of a membrane occupying a region $\Omega$. It is modelled by an eigenvalue equation (cf. also Neumann eigenvalue; Rayleigh–Faber–Krahn inequality). Let $- T \Delta w ( x , y )$ be denoted by $A w$, and $\rho ( x , y ) w ( x , y )$ by $B w$. $\Delta$ is the Laplace operator $( \partial ^ { 2 } / \partial x ^ { 2 } + \partial ^ { 2 } / \partial y ^ { 2 } )$. The deflection $w ( x , y ) = 0$ on the boundary $\partial \Omega$ of $\Omega$ for all $t > 0$. (Here, $T$ denotes the uniform membrane tension, $\rho$ is mass per unit length.) Then $A w = \lambda B w$ is the disguised equation of motion, with eigenvalue $\lambda = \omega ^ { 2 }$, where $\omega$ is the natural frequency of vibration.

One can introduce the following product for arbitrary functions satisfying boundary conditions whose gradients are square integrable in $\Omega$:

\begin{equation*} \{ w , v \} = \int \int _ { \Omega } [ A w ( x , y ) ] v ( x , y ) d x d y = \end{equation*}

\begin{equation*} = \int \int _ { \Omega } w ( x , y ) [ A v ( x , y ) ] d x d y. \end{equation*}

Putting $v = w$, one obtains an energy equation for a freely vibrating membrane. It connects the two basic energy forms: potential and kinetic. Rayleigh's principle relates the value of the smallest (fundamental) natural frequency of the system to the minimum, attained over all possible forms of vibration, of the ratio of the average kinetic energy over average potential energy, computed over a single cycle of vibration.

Note that $A$ being self-adjoint implies conservation of energy. So the problem with self-adjoint operators is not very realistic: The vibrating string does not know how to stop vibrating. It will go on forever with the same frequency and the same amplitude. However, a real string will insist on dissipating some of its energy and this has to be reflected in the properties of the operator.

Generally, a correction is made by inserting first-order differential terms into the differential equation, but not always.

In the example of a vibrating elastic shaft one has (with suitable boundary conditions) a self-adjoint Euler–Bernoulli equation:

\begin{equation*} L u = \frac { \partial ^ { 2 } } { \partial x ^ { 2 } } \left( E I ( x ) \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } \right) + \rho A ( x ) \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }. \end{equation*}

S. Timoshenko suggested a fifth-order derivative correction term: $G ( x ) \partial ^ { 5 } /\partial x ^ { 4 } \partial t$ or similar, taking care of small dissipative effects caused by rotational inertia deforming the shape of the cross-sectional area. With this term, and also possible first-order derivative terms included, the operator $L$ is no longer self-adjoint, and the eigenvalues become complex numbers. Ignoring small damping terms, the equation

\begin{equation*} ( L - \operatorname { Re } ( \lambda I ) u = f \end{equation*}

can be solved. Here $\lambda$ is an eigenvalue of $L$, and $\operatorname { Re } ( \lambda )$ is the real part of $\lambda$. Then the approximate dissipation-free solution can be written as $\hat { u } = ( L - \operatorname { Re } ( \lambda ) I ) ^ { - 1 } f$. Observe that the indicated inverse exists, since $\operatorname { Re } ( \lambda )$ is not an eigenvalue. If $f$ is close to an eigenfunction corresponding to the "true" $\lambda$ and $\operatorname { Re } ( \lambda )$ is very close to the pole of the resolvent, the response $\widehat{u}$ may become very large. This is a classical example of natural frequency resonance.

In the energy-conserving problems described above, the domain of the operator $L$ is compact, the inverse $L ^ { - 1 }$ is a compact operator, the spectrum of $L$ consists of real eigenvalues only, and the only accumulation point for the eigenvalues is at infinity.

Complications arise in quantum physics, where, in general, the domain of the operator is not compact and a continuous spectrum is superimposed on the true eigenvalues. Consider the Schrödinger operator $- h \Delta + V ( x )$, where $V$ is a potential (cf. also Schrödinger equation). Since boundary values are absent, the spectrum of $- h \Delta$ is the positive part of the real line. Since solutions cannot be contained in a compact set, barriers set by the potential which produce a well of minimum energy surrounded by "hills" are not respected. In fact, there is a unique (meromorphic) continuation of the resolvent, whereby the solutions tunnel through the obstacles. This refutes the classical laws of physics, under which particles can be trapped at the bottom of a potential well (corresponding to a minimal energy level).

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