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Let a homogeneous plate $0<x<L$ begin at uniform temperature $T_0$, cool with zero-temperature Dirichlet boundary conditions until time $t_b$, then switch to insulated Neumann boundary conditions. For thermal diffusivity $\kappa >0$, the Fourier sine series during cooling is $T=(4T_0/\pi ){\sum }_{n\ge 1,n\text{ odd}}\sin (n\pi x/L)e^{-\kappa (n\pi /L{)}^{2}t}/n$. Integrating the insulated heat equation shows that its integral average is conserved. All nonconstant Fourier cosine series modes then decay, proving that the final uniform temperature is $T_{\infty }=(8T_0/{\pi }^2){\sum }_{n\ge 1,n\text{ odd}}{e}^{-\kappa (n\pi /L{)}^2{t}_b}/n^2$. Its limits are $T_0$ at zero bath time and zero at infinite bath time. Instantaneous changes of boundary condition can create initial-boundary corner incompatibilities without invalidating the positive-time solution.

Subtract a known function with the required boundary trace so that the remaining unknown has homogeneous boundary conditions. For an interval with Dirichlet boundary conditions $g(t),h(t)$, $\ell =(1-x/L)g+(x/L)h$ is a simple lifting. In $w_t=w_{xx}-cw$, $z=w-\ell$ has forcing $-{\ell }_t-c\ell$ and zero endpoint values. This improves Fourier sine series convergence.

For $y_{tt}-y_{xx}=F(t)\delta (x-x_0)$ with zero endpoint displacement, a Fourier sine series gives $q_n^{\prime \prime }+k_n^2{q}_n=2F(t)\sin (k_n{x}_0)/L$. With the $e^{-i\omega t}$ Fourier transform convention, the causal inverse of this oscillator uses the displayed denominator. Its inverse transform is $H(t)\sin (k_{n}t)/k_n$, giving $q_n(t)=2\sin (k_n{x}_0)/(L{k}_n){\int }_{-\infty }^{t}F(s)\sin [k_n(t-s)]ds$ when the past integral exists. The plain denominator $k_n^2-{\omega }^2$ requires a pole prescription: its Cauchy principal value gives the time-symmetric kernel $\sin (k_n|t|)/(2k_n)$ instead. Homogeneous sine and cosine modes are additional freedom unless temporal data or causality are specified.

On $0<x<L_x$, $0<y<L_y$ with positive side lengths and zero Dirichlet boundary conditions, separation of variables for $-\Delta u=\lambda u$ giveswith the displayed eigenvalues. The one-dimensional endpoint conditions exclude constant and hyperbolic factors and force the positive integer sine frequencies. The product Fourier sine series basis is complete in the square-integrable functions on the rectangle. Consequently an eigenfunction with prescribed $\lambda$ is a linear combination of exactly those modes with ${\lambda }_{mn}=\lambda$; if none match, only the zero solution exists. Distinct pairs with the same sum give an eigenvalue with geometric multiplicity greater than one. On a square of side $\pi$, for instance, $\lambda =25$ has precisely the pairs $(3,4)$ and $(4,3)$, so its eigenspace has dimension two.

On the strip $\mathbb{R}\times (0,a)$, with $a>0$, choose the convention $\Delta G=\delta$ and zero Dirichlet boundary conditions on both horizontal sides. The Dirichlet Green function isTo derive the coefficient, use the Fourier sine series completeness relation $\delta (y-c)=(2/a){\sum }_{n\ge 1}\sin (n\pi y/a)\sin (n\pi c/a)$. Each horizontal mode solves $({\partial }_x^2-{\kappa }_n^2)g_n=\delta (x-b)$ with ${\kappa }_n=n\pi /a$. Its decaying Green function is $-e^{-{\kappa }_n|x-b|}/(2{\kappa }_n)$, since the first derivative has jump one. Multiplying by $2/a$ gives the displayed coefficient.

The series converges absolutely when $x\ne b$, vanishes on $y=0,a$, and decays as $|x-b|\to \infty$. At the point source $(b,c)$ it has the expected logarithmic singularity; the source equation is interpreted in the distribution sense. The strip width sets the longest horizontal decay scale $a/\pi$.

For an integrable function on $[-L,L]$, with $L>0$ and integer $n\ge 1$, the displayed Fourier coefficient multiplies the mode $\sin (n\pi x/L)$ in its real Fourier series. Fourier orthogonality fixes the normalization because the squared mode has integral $L$. If $f$ is odd, the product with the sine is even, so $b_n=(2/L){\int }_0^{L}f(x)\sin (n\pi x/L)dx$. These are also the coefficients of the Fourier sine series of the restriction to $(0,L)$.

Extend $F_L(x)=x(L-|x|)$ from $[-L,L]$ to a periodic function of period $2L$, with $L>0$. It is odd and continuous, so its Fourier series contains only sine terms. Its Fourier coefficients areTwo applications of integration by parts prove the coefficient formula. The first has no endpoint contribution because $x(L-x)$ vanishes at both endpoints; the second leaves a multiple of $1-\cos (n\pi )$. Thus only odd frequencies survive, giving the displayed Fourier sine series. Its absolutely summable coefficients give uniform convergence, and Fourier orthogonality identifies its sum with $F_L$.

Taking $L=1$, ${\int }_{-1}^1{F}_1(x{)}^{2}dx=2{\int }_0^1{x}^2(1-x{)}^{2}dx=1/15$. Parseval's identity therefore gives

On $0\le x\le L$, the Fourier sine series of ${\sin }^2(\pi x/L)$ has zero even-index coefficients and odd-index coefficients $b_n=8/[\pi n(4-n^2)]$. Substitute $u=\pi x/L$ in the coefficient integral and use ${\sin }^{2}u\sin (nu)=\frac{1}{2}\sin (nu)-\frac{1}{4}\sin ((n+2)u)-\frac{1}{4}\sin ((n-2)u)$. For odd $n$, integrating gives $b_n=(2/\pi )[1/n-\frac{1}{2}(1/(n+2)+1/(n-2))]$; reflection about the midpoint makes the even coefficients zero. The $O(n^{-3})$ decay gives absolute and uniform convergence, and Fourier orthogonality identifies the sum. In the heat equation with zero endpoint temperatures, multiply each coefficient by $\exp [-\kappa (n\pi /L{)}^{2}t]$. With insulated endpoints, the same initial profile is instead just the constant and second cosine modes of a Fourier cosine series.

On $0\le x\le L$, with $L>0$, the tent $f(x)=\min (x,L-x)$ has Fourier sine series coefficients $b_n=4L\sin (n\pi /2)/({\pi }^2{n}^2)$. Split the coefficient integral at $L/2$ and integrate its two linear pieces by parts; their cosine endpoint terms cancel and their sine terms add. Reflection about the midpoint makes every even-index coefficient zero, and the remaining odd-index coefficients alternate in sign. The $n^{-2}$ bound gives absolute convergence and uniform convergence; the series equals the continuous tent, including its zero endpoints.

For a unidirectional incompressible flow between stationary plates a distance $H>0$ apart, removing the driving pressure gradient leaves $u_t=\nu u_{yy}$ when the kinematic viscosity $\nu >0$ is constant. The no-slip boundary condition gives $u(0,t)=u(H,t)=0$. For an initial profile $g$ in L2 space, separation of variables and the Fourier sine series give the displayed solution, withEach normal mode decays at a rate proportional to $n^2$. The lowest mode present dominates at late times; when $b_1\ne 0$, its time constant is $H^2/(\nu {\pi }^2)$, consistent with the viscous diffusion time. The solution approaches the initial data in L2 space; compatible smooth profiles also give the familiar pointwise initial condition. This is a heat equation for momentum diffusion, rather than temperature.

On $0<x<a$, $0<y<b$, the bounded harmonic function with value $F$ on the open upper edge and zero on the other three open edges has the sine-hyperbolic series in the title. Separation of variables gives the modes, and the Fourier sine series of a constant selects only odd $n$. The series converges exponentially on compact subsets of the interior, justifying its derivatives in Laplace's equation.

For $F\ne 0$, the upper corners have incompatible one-sided boundary limits, so the solution cannot extend continuously to them. Assigning isolated corner values does not alter the bounded interior Dirichlet problem. For numerical evaluation, the stable mode ratio isThis avoids overflow from separately evaluating large hyperbolic sines.

On $0<x<L$ with zero-temperature endpoints, let the initial temperature rise linearly from zero to height $H$ at $L/2$ and then fall linearly to zero. Integration by parts on the two halves gives Fourier sine series coefficients $8H\sin (n\pi /2)/({\pi }^2{n}^2)$. Under the heat equation $T_t=D{T}_{xx}$ with $D>0$, each coefficient decays by $e^{-D(n\pi /L{)}^{2}t}$, giving the displayed solution. Only odd spatial modes occur; absolute coefficient summability proves uniform convergence for $t\ge 0$ via the Weierstrass M-test. This spatial profile also occurs in a midpoint-plucked fixed-end string, where the time factors oscillate instead of decaying.

On the unit interval $0<x<1$, a fixed-end string with unit wave speed and damping coefficient1 obeysExpanding in the Fourier sine series $y={\sum }_{n\ge 1}{T}_n(t)\sin (n\pi x)$ gives $T_n^{\prime \prime }+T_n^{\prime }+n^2{\pi }^2{T}_n=0$. ThusAll these modes are underdamped; their frequency depends on the endpoint geometry as well as the damping. Nonunit length, speed or damping changes this modal equation.

For a sufficiently regular finite-energy solution, the energy dissipation identity for a linearly damped string isMultiply the equation by $y_t$ and integrate by parts; the fixed endpoints remove the boundary term. This identifies exactly where mechanical energy is lost. The modal solution also separates decay of the amplitude envelope from oscillatory changes of the actual displacement.

For $x>0$, write $t=\log x$ and $y=xv(t)$. The chain rule gives $y^{\prime }=v+v_t$, $y^{\prime \prime }=x^{-1}(v_t+v_{tt})$, and thereforeOn $1<x<e^{\ell }$ with homogeneous Dirichlet boundary conditions, the eigenfunctions are $x\sin (n\pi \log x/\ell )$ and the eigenvalues are $-(n\pi /\ell {)}^2$. The weighted inner product with weight $x^{-3}$ becomes ${\int }_0^{\ell }\overline{v}wdt$, so their squared norms are $\ell /2$ and their orthogonality is exactly that of the Fourier sine series. The weight is the factor that makes this conjugacy preserve the inner product.

A string of length $L$ with fixed endpoints, initially displaced into a triangular tent of height $k$ at its midpoint and released from rest, has a Fourier sine series with the coefficients in the title. Integrating the two linear halves gives $a_n$, and symmetry cancels the even modes. Each coefficient evolves as $a_n\cos ({\omega }_{n}t)$ under the wave equation. The displacement vanishes everywhere at $t=L/(2c)$, reverses sign at $t=L/c$, and returns after $2L/c$. A straight configuration at an intermediate instant does not mean the string is at rest: its kinetic energy is then nonzero. The initial slope corner gives a finite-energy weak solution whose corner lines propagate and reflect at the ends.

Expand a real function with zero endpoint values in its Fourier sine series, $y={\sum }_{n\ge 1}{a}_n\sin (n\pi x/L)$. The Parseval identity gives $\int y^2=(L/2)\sum a_n^2$. For its weak derivative, the Fourier cosine series has coefficients $(n\pi /L)a_n$ and zero constant term, soThe derivative coefficient identity follows by integration by parts, rather than by assuming pointwise differentiation of an arbitrary series. This proves the Wirtinger inequality and its equality case. Under unit L2 norm, the only real minimizers are $y=\pm \sqrt{2/L}\sin (\pi x/L)$.

A normal mode, also called an eigenmode, is a pattern of free oscillation in a linear system whose relative displacements stay fixed while a single temporal factor changes their size and sign. For an undamped stable vibrational mode,Here $\varphi$ is the mode shape, $A$ sets the amplitude, $\omega$ is the eigenfrequency expressed as an angular frequency, and $\phi$ is the oscillation phase. Every snapshot is a scalar multiple of the same mode shape; neighbouring regions may move in opposite directions. Normal modes of a double pendulum show that normal modes also describe coupled discrete systems, beyond strings. Their shapes are eigenvectors, rather than spatial eigenfunctions. ^[ref]

For the ideal wave equation on a string, take a uniform string of length $L$ with constant tension $\tau$ and linear mass density $\mu$. Assume small transverse slope, negligible bending stiffness, no damping, and fixed endpoints. Its wave speed is $c=\sqrt{\tau /\mu }$, and the nonzero normal modes areThe angular frequency ${\omega }_n$ is in radians per second; the frequency $f_n$ is in hertz. The wavelength is $2L/n$. The mode shape has $n$ loops separated by nodes of a standing wave, including both fixed endpoints. ^[ref]

As a worked example, choose $L=1\mathrm{m}$, $\tau =1\mathrm{N}$, and $\mu =1\mathrm{kg}{\mathrm{m}}^{-1}$. Then $c=1\mathrm{m}{\mathrm{s}}^{-1}$ and $f_1=0.5\mathrm{Hz}$, $f_2=1\mathrm{Hz}$, $f_3=1.5\mathrm{Hz}$. Their oscillation periods are $T_1=2\mathrm{s}$, $T_2=1\mathrm{s}$, and $T_3=2/3\mathrm{s}$. These are deliberately simple model parameters, rather than measurements of a musical instrument. The plot divides displacement by $A$ so that the three mode shapes can be compared, and uses each mode's own oscillation period for its snapshots.

For $\phi =0$, at $t=T_n/4$ every point has zero displacement, but most points have nonzero velocity. Only zeros of the mode shape are permanent nodes of a standing wave. At $t=T_n/2$ the original mode shape has reversed sign, and at $t=T_n$ it returns. The synchronized conversion between kinetic energy and potential energy is shown in normal-mode energy exchange.

The short derivation uses separation of variables: substituting $u=\varphi q$ into the wave equation gives ${\varphi }^{\prime \prime }+k^2\varphi =0$ and $q^{\prime \prime }+c^2{k}^{2}q=0$. The fixed-end boundary conditions remove the cosine spatial term and require $\sin (kL)=0$, hence $k=n\pi /L$ and ${\omega }_n=ck$. Different geometries and boundary conditions give different eigenfunctions; a sine mode shape is a property of this uniform fixed string, not a universal form of an eigenmode.

The usefulness is that a complicated vibration becomes a collection of harmonic oscillators. For sufficiently regular fixed-end initial data, the Fourier sine series givesThe initial displacement and velocity determine $a_n$ and $b_n$. This is the superposition principle in a Fourier sine basis, and modal energy distribution measures how strongly each normal mode is excited. The striking simplification is that each component keeps its identity while their sum can continually change shape. ^[ref]

These independent normal modes belong to the linear model. At larger slopes, changing tension introduces nonlinear effects and can couple the modes. A linearly damped string still admits a Fourier sine basis, but its temporal factors decay and need not oscillate; its energy dissipation identity for a linearly damped string replaces conservation of energy. More general damping need not leave undamped mode shapes independent.

Given a function on $(0,L)$, reflect it oddly onto $(-L,0)$ and repeat with period $2L$. Nonzero endpoint limits produce jumps at the joins. The resulting Fourier series is the Fourier sine series of the original function; its value at a jump converges to the mean of the one-sided limits under the Dirichlet convergence theorem for Fourier series.

Project the expansion onto one normalized basis function. Orthogonality removes every other coefficient:Consequently,This is a Fourier sine series in normalized form. For a square-integrable $f$, completeness gives convergence in the norm of the function space; pointwise convergence requires the usual additional regularity. If $f$ is a normalized wavefunction, the Born rule gives probabilities $|a_n{|}^2$ and ${\sum }_n|a_n{|}^2=1$.

The useful change of independent variable in a second-order ODE is $t=\log x$, followed by $y(x)=xv(t)$. The chain rule givesThus the eigenvalue equation becomes $v_{tt}=\lambda v$ on $0<t<1$, and the endpoint values become $v(0)=v(1)=0$. At the specified eigenvalues,The first endpoint forces $A=0$; the second holds for every positive integer $k$. Choosing a convenient normalization yieldsEach nonzero constant multiple is the same eigenfunction up to normalization. This logarithmic conjugacy of a repeated-root Euler operator turns the Cauchy-Euler equation into the familiar Fourier sine series problem on a unit interval.