Why do we need the Schrödinger equation if we have a wave equation? Is the Schrödinger equation a wave equation?  Why do we need the Schrödinger equation if earlier we have a wave equation?
Also, could you explain the difference between these two equations?
 A: TL; DR

*

*Mathematically the two equations do not belong to the same type

*There is a lot of specific physical content associated with the Schrödinger equation

In purely mathematical terms, the second order partial differential equations are classified into elliptic, parabolic and hyperbolic equations. The most common representatives of these three classes occurring in physics are

*

*Laplace equation
$$\nabla^2 u(x,y,z) = 0$$

*Diffusion equation
$$D\nabla^2 u(x,y,z,t) = \frac{\partial u(x,y,z,t)}{\partial t}$$

*Wave equation
$$\nabla^2 u(x,y,z,t) = \frac{1}{v^2}\frac{\partial^2 u(x,y,z,t)}{\partial t^2}$$
Although the there names originate from particular applications, they are frequently used  to denote particular mathematical type of equation. As you see the mathematical classification here is based on the type of derivatives involved (parabolic/diffusion equation has a first order derivative) and the sign of the second order derivatives (Laplace equation has all the derivatives of the same sign, whereas the wave equation has one derivative with a different sign.)
Then there are even more domain-specific names, which typically imply certain type of coefficients and/or certain type of inhomogeneous terms. Thus, Poisson equation is an inhomogeneous Laplace equation, whereas Schrödinger equation is a Diffusion equation, with a complex diffusion coefficient, often with a term with zero derivative (the potential term) and potentially also the first derivative term (in magnetic field). Moreover, in some settings Schrödinger equation may contain higher order derivatives, be formulated for multicomponent functions (e.g., in presence of spin), and even for descrete rather than continuous functions (although continuous in time). In other words, it is mathematically and physically very different from the wave equation.
A: It's not actually a "wave equation." A wave equation in 1D looks like:
$$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}$$
Two derivatives in time, two derivatives in position.
The Schodinger Equation, on the other hand has the form:
$$i \hbar \frac{\partial}{\partial t} \Psi(x, t)=\left[-\frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}}+V(x, t)\right] \Psi(x, t) .$$
which for no potential is:
$$i \hbar \frac{\partial}{\partial t} \Psi(x, t)=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} \Psi(x, t) .$$
This has only one derivative in time, not two. Also, the solution is a complex number, while the wave equation is real.
Why is it called the "wave equation"? Well, because it moves around similar to a wave enough that people thought of it intuitively as a wave. The reality though is that this equation is actually the heat equation with complex coefficients, so these "waves of probability" aren't so much waves as they are functions that move in the same way heat propagates (not exactly, because there are complex numbers added in).
A: Quoting Wikipedia

The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system.

It describes the evolution of a state vector $|\psi(t)\rangle $ under the given Hamiltonian.
$$i\hbar \frac{d}{dt}|\psi(t)\rangle =\hat{H}|\psi(t)\rangle $$

On the other hand, Again Quoting Wikipedia

The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.

$$\nabla^2\psi(x,t)=\frac{1}{c^2}\frac{\partial^2 \psi(x,t)}{\partial t^2}$$


Is Schrodinger a wave equation?

It's not identical to the classical wave equation. You can rather say it's a probability wave equation if you like.

Why do we need the Schrodinger equation if earlier we have a wave equation?

The classical wave equation doesn't describe the evolution of the quantum state vector.

Also explain the difference between these two equations?

There isn't anything.
Edit: There are comments on the fact there is something (perhaps many things) that are common in the two. Well, they are right. I'm not considering here, mathematical similarities but with physical. Some authors (See Quantum Mechanics L. Shiff) try to derive Schrodinger's equation from the classical wave equation and using the de-Broglie hypothesis. But I find it confusing, I prefer to look at it rather as a postulate of Quantum mechanics that is independent of the classical wave equation.

~Reference

*

*Schrodinger Equation

*Wave Equation
A: The difference between the two is more vague than one may think. Shroedinger himself in his 1926 paper introduced his equation as a second order time derivative equation. The way he wrote his equation was rather like this:
$$
\partial_t^2 \psi = - \hat{H}^2 \psi.\tag{1}
$$
Here $\hat{H}$ is the usual Hamiltonian $\hat{H}=\frac{\hat{p}^2}{2 m}+V(r)$, and wave-function $\psi$ was assumed to be real. The complex, first order version of this equation, $$
i\partial_t \psi = \hat{H}\psi,\tag{2}
$$
was rather a mathematical trick to simplify the solution. One can notice, that any $\psi$ satisfying Eq. (2) would also satisfy Eq. (1). The second solution to the Eq. (1) could be obtained by solving equation
$$
-i\partial_t \psi = \hat{H}\psi,\tag{3}
$$
which is just a complex conjugate to Eq. (2). It doesn't tell us much new about the system, but allows to make the solution of Eq. (1) real, as was initially intended.
So, how did we decide that the complex first order equation is the true equation, and the real second order equation is the wrong one? I am not sure and would like to know myself. I was told at school that people really liked that probability in the complex Shroedinger equation was expressed as $|\psi|^2$ (Born, 1926), while the second order real equation (1) lacked such interpretation and would require time derivatives of $\psi$ to define probability. I am not sure this is a real reason, but I can see why people would want to stick to something simpler and with better interpretation.
A: This can sound a bit confusing, but one very useful way to put things is that it is important to distinguish between "a" wave equation and "the" wave equation.

*

*The wave equation is a very specific partial differential equation (PDE), in two or more dimensions (one time dimension plus one or more space dimensions), which has the highly-codified form $$\frac{\partial^2 f}{\partial t^2}-\frac{1}{c^2}\frac{\partial^2 f}{\partial x^2} = 0 \tag1$$ in one spatial dimension (and $\frac{\partial^2 f}{\partial t^2}-\frac1{c^2}\nabla^2f=0$ in higher spatial dimensions, where $\nabla^2$ is the laplacian operator). This PDE governs, among other things, the propagation of light and sound, as well as things like e.g. seismic waves.


*However, the term "wave equation" can also be used in a more generic way. In this usage, the term can denote any partial differential equation which governs the evolution of a wave phenomenon. This is a broad category, with fuzzy edges, and it is hard to define $-$ largely because it is hard to provide a hard, objective definition of what "wave phenomenon" means.
This category includes a large array of equations, which stretch from generalizations of 'the' wave equation $(1)$ (to do things like spatial variations of the refractive index of the medium), through the inclusion of nonlinearities (which covers a wide range of wave phenomena), to fancier and more complex PDEs (with the Korteweg–De Vries equation being a particularly useful example) that still model wave phenomena.
As far as this separation goes, the Schrödinger equation is obviously different from the wave equation, but it is still a wave equation.
Indeed, this is precisely how it was found. When Louis de Broglie postulated his hypothesis that matter particles have wave properties and that the wavelength corresponds to the particle's momentum (via $\lambda=h/p$), the very first question that this raised was

if quantum particles are a wave phenomenon, what wave equation governs them?

The answer to this question could have been 'the' wave equation, but it could also have been any one from the broader class of wave equations. After de Broglie's work, there was an active search for what that wave equation might be, and this is the question that Schrödinger answered.
So, what was the answer? The Schrödinger equation shares some features with the wave equation $(1)$, but it is also very different. In its most basic form, in one spatial dimension, it reads
$$
i\hbar \frac{\partial \psi}{\partial t} + \frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2} - V(x) \psi = 0.
\tag 2
$$
Like the wave equation $(1)$, it is a partial differential equation in two variables, $x$ and $t$, and it is linear, and second-order in $x$. However, there are important differences:

*

*The Schrödinger equation is only of first order in $t$. This means that it is of parabolic type (as opposed to the wave equation, which is of hyperbolic type), and this has strong effects on the mathematical structure and properties of the equation.

*The Schrödinger equation is explicitly complex-valued, which forces the wavefunction to be complex-valued as well.

*The Schrödinger equation includes a term, $V(x)\psi(x)$, proportional to the potential energy $V(x)$ of the particle in the corresponding problem in classical mechanics.

That final point is absolutely key: the wave equation $(1)$ is completely fixed (there is one and only one such equation), but the Schrödinger equation will change depending on what $V(x)$ is. This means that each quantum system (corresponding to a suitable classical system) will have its very own Schrödinger equation, with different dynamics, and this is precisely what you need: the Schrödinger equation provides the right amount of flexibility to describe different systems with different properties and different behaviours.
