Normal mode in the context of "Harmonic series (music)"

In physics, magnetosonic waves, also known as magnetoacoustic waves, are low-frequency compressive waves driven by mutual interaction between an electrically conducting fluid and a magnetic field. They are associated with compression and rarefaction of both the fluid and the magnetic field, as well as with an effective tension that acts to straighten bent magnetic field lines. The properties of magnetosonic waves are highly dependent on the angle between the wavevector and the equilibrium magnetic field and on the relative importance of fluid and magnetic processes in the medium. They only propagate with frequencies much smaller than the ion cyclotron or ion plasma frequencies of the medium, and they are nondispersive at small amplitudes.

There are two types of magnetosonic waves, fast magnetosonic waves and slow magnetosonic waves, which—together with Alfvén waves—are the normal modes of ideal magnetohydro­dynamics. The fast and slow modes are distinguished by magnetic and gas pressure oscillations that are either in-phase or anti-phase, respectively. This results in the phase velocity of any given fast mode always being greater than or equal to that of any slow mode in the same medium, among other differences.

A two-dimensional elastic membrane under tension can support transverse vibrations. The properties of an idealized drumhead can be modeled by the vibrations of a circular membrane of uniform thickness, attached to a rigid frame. Based on the applied boundary condition, at certain vibration frequencies, its natural frequencies, the surface moves in a characteristic pattern of standing waves. This is called a normal mode. A membrane has an infinite number of these normal modes, starting with a lowest frequency one called the fundamental frequency.

There exist infinitely many ways in which a membrane can vibrate, each depending on the shape of the membrane at some initial time, and the transverse velocity of each point on the membrane at that time. The vibrations of the membrane are given by the solutions of the two-dimensional wave equation with Dirichlet boundary conditions which represent the constraint of the frame. It can be shown that any arbitrarily complex vibration of the membrane can be decomposed into a possibly infinite series of the membrane's normal modes. This is analogous to the decomposition of a time signal into a Fourier series.

Planar transmission lines are transmission lines with conductors, or in some cases dielectric (insulating) strips, that are flat, ribbon-shaped lines. They are used to interconnect components on printed circuits and integrated circuits working at microwave frequencies because the planar type fits in well with the manufacturing methods for these components. Transmission lines are more than simply interconnections. With simple interconnections, the propagation of the electromagnetic wave along the wire is fast enough to be considered instantaneous, and the voltages at each end of the wire can be considered identical. If the wire is longer than a large fraction of a wavelength (one tenth is often used as a rule of thumb), these assumptions are no longer true and transmission line theory must be used instead. With transmission lines, the geometry of the line is precisely controlled (in most cases, the cross-section is kept constant along the length) so that its electrical behaviour is highly predictable. At lower frequencies, these considerations are only necessary for the cables connecting different pieces of equipment, but at microwave frequencies the distance at which transmission line theory becomes necessary is measured in millimetres. Hence, transmission lines are needed within circuits.

The earliest type of planar transmission line was conceived during World War II by Robert M. Barrett. It is known as stripline, and is one of the four main types in modern use, along with microstrip, suspended stripline, and coplanar waveguide. All four of these types consist of a pair of conductors (although in three of them, one of these conductors is the ground plane). Consequently, they have a dominant mode of transmission (the mode is the field pattern of the electromagnetic wave) that is identical, or near-identical, to the mode found in a pair of wires. Other planar types of transmission line, such as slotline, finline, and imageline, transmit along a strip of dielectric, and substrate-integrated waveguide forms a dielectric waveguide within the substrate with rows of posts. These types cannot support the same mode as a pair of wires, and consequently they have different transmission properties. Many of these types have a narrower bandwidth and in general produce more signal distortion than pairs of conductors. Their advantages depend on the exact types being compared, but can include low loss and a better range of characteristic impedance.

Multi-mode optical fiber is a type of optical fiber mostly used for communication over short distances, such as within a building or on a campus. Multi-mode links can be used for data rates up to 800 Gbit/s. Multi-mode fiber has a fairly large core diameter that enables multiple light modes to be propagated and limits the maximum length of a transmission link because of modal dispersion. The standard G.651.1 defines the most widely used forms of multi-mode optical fiber.

A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical vibrational frequencies range from less than 10 Hz to approximately 10 Hz, corresponding to wavenumbers of approximately 300 to 3000 cm and wavelengths of approximately 30 to 3 μm.

Vibrations of polyatomic molecules are described in terms of normal modes, which are independent of each other, but each normal mode involves simultaneous vibrations of parts of the molecule. In general, a non-linear molecule with N atoms has 3N − 6 normal modes of vibration, but a linear molecule has 3N − 5 modes, because rotation about the molecular axis cannot be observed. A diatomic molecule has one normal mode of vibration, since it can only stretch or compress the single bond.

In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.

Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems, in the linear case, involves the eigenfunctions of a differential operator.