Question

The Klein-Gordon equation is \(\nabla^{2} u=\left(1 / v^{2}\right) \partial^{2} u / \partial t^{2}+\lambda^{2} u .\) This equation is of interest in quantum mechanics, but it also has a simpler application. It describes, for example, the vibration of a stretched string which is embedded in an elastic medium. Separate the one-dimensional Klein-Gordon equation and find the characteristic frequencies of such a string.

Short answer

The characteristic frequencies of the string are given by \(\omega = v \sqrt{k^{2} + \lambda^{2}}\).

Step-by-step solution

Write the one-dimensional Klein-Gordon equation

Given the Klein-Gordon equation: \[abla^{2} u = \left( \frac{1}{v^{2}} \right) \frac{\partial^{2}u}{\partial t^{2}} + \lambda^{2}u \]For the one-dimensional case, the Laplacian \(abla^{2}\) reduces to:\[\frac{\partial^{2}u}{\partial x^{2}} = \left( \frac{1}{v^{2}} \right) \frac{\partial^{2}u}{\partial t^{2}} + \lambda^{2}u \]

Assume a solution of the form

Assume a separable solution of the form:\[u(x,t) = X(x)T(t)\]Substitute this assumed solution into the one-dimensional Klein-Gordon equation.

Substitute and separate variables

Substitute the assumed solution into the one-dimensional Klein-Gordon equation:\[\frac{\partial^{2}}{\partial x^{2}}[X(x)T(t)] = \left( \frac{1}{v^{2}} \right) \frac{\partial^{2}}{\partial t^{2}}[X(x)T(t)] + \lambda^{2}X(x)T(t)\]This results in:\[T(t) \frac{d^{2} X(x)}{d x^{2}} = \left( \frac{X(x)}{v^{2}} \right) \frac{d^{2} T(t)}{d t^{2}} + \lambda^{2}X(x)T(t)\]

Divide by product solution and separate

Divide through by \(X(x)T(t)\) to obtain:\[\frac{1}{X(x)} \frac{d^{2} X(x)}{d x^{2}} = \frac{1}{v^{2}T(t)} \frac{d^{2} T(t)}{d t^{2}} + \lambda^{2}\]Since the left side depends only on \(x\) and the right side depends only on \(t\), both sides must equal a separation constant which we will call \( -k^{2} \) for convenience: \[\frac{1}{X(x)} \frac{d^{2} X(x)}{d x^{2}} = -k^{2}\]\[\frac{1}{v^{2}T(t)} \frac{d^{2} T(t)}{d t^{2}} = -k^{2} - \lambda^{2}\]

Solve the spatial part

Solve the equation for \(X(x)\): \[\frac{d^{2} X(x)}{d x^{2}} + k^{2} X(x) = 0\]Assume solutions of the form \(X(x) = A \cos(kx) + B \sin(kx)\), which are standard solutions for such second-order differential equations.

Solve the temporal part

Solve the equation for \(T(t)\): \[\frac{d^{2} T(t)}{d t^{2}} + v^{2}(k^{2} + \lambda^{2})T(t) = 0\]Assume solutions of the form \(T(t) = C \cos(\omega t) + D \sin(\omega t)\), where:\[\omega^{2} = v^{2}(k^{2} + \lambda^{2})\]

Find the characteristic frequencies

The general solution for time-dependent part involves frequencies \(\omega\), which are given by:\[\omega = v \sqrt{k^{2} + \lambda^{2}}\]Here, \(k\) are the wavenumbers representing the spatial frequency components, and \(\lambda\) represents a constant related to the medium properties.

Key concepts

Quantum Mechanics

The Klein-Gordon equation originally arises in the context of quantum mechanics. In this field, it describes particles like mesons that obey relativistic mechanics rather than classical mechanics. The equation is a relativistic version of the Schrödinger equation, which you might already know describes the wave function of a quantum particle. Unlike the Schrödinger equation, which is first-order in time, the Klein-Gordon equation is second-order in time. This makes it analogous to classical wave equations, but it incorporates relativistic principles. Understanding this equation provides a bridge that connects the classical theory of vibrations and waves to the more complex theories needed for the particles moving at close to light speed. Such an understanding is crucial for higher-level studies in quantum field theory, where the Klein-Gordon equation is one of the simplest field equations you will encounter.

Vibration of a String

The Klein-Gordon equation can also describe the vibration of a string. Imagine a string that is embedded in an elastic medium. This situation is more intuitive and provides a physical example of the abstract mathematical concepts. When you pluck this string, it vibrates in patterns that depend on boundary conditions and properties of the string and medium. With the equation, you can model these vibrations and predict how the string moves over time. The equation shows how the displacement at any point on the string evolves. By solving it, you can understand the different modes of vibration, which refer to the characteristic ways the string can oscillate. These modes correspond to different frequencies, much like how a guitar string produces different notes depending on how it is plucked and where it is fretted.

Separation of Variables

To solve the Klein-Gordon equation, we used a method called separation of variables. This method involves assuming that the solution can be written as a product of functions, each depending only on one variable. For the vibration of the string, we assumed a solution of the form: \(u(x,t) = X(x)T(t)\). By substituting this form into the Klein-Gordon equation, the problem converts into two separate differential equations: one for space and one for time. This technique simplifies the original partial differential equation into simpler, ordinary differential equations. Each of these can often be solved more easily. The solutions to these simpler equations are then multiplied together to form the complete solution. Separation of variables is a powerful tool and is widely used in solving partial differential equations across physics and engineering.

Characteristic Frequencies

By solving the separate equations in the separation of variables method, you uncover the characteristic frequencies of the vibrating system. These frequencies are intrinsic to the system and depend on properties like tension, density, and the inherent elasticity of the medium. The frequencies at which the system naturally oscillates are called its natural frequencies or eigenfrequencies. For the Klein-Gordon equation, these frequencies are given by: \(\omega = v \sqrt{k^{2} + \lambda^{2}}\). Here, \(\omega\) represents the angular frequency, \(v\) is the velocity of wave propagation, \(k\) is the wavenumber corresponding to spatial frequency, and \(\lambda\) is a constant representing the medium's property. These characteristic frequencies are crucial in understanding resonance and stability in physical systems. If an external force matches one of these frequencies, the system can resonate, leading to large amplitude oscillations. This concept is vital in designing structures and understanding phenomena like musical instruments, seismic waves, and more.