2 水波算法 Water Ripple Simulation

1 波源产生 Water Ripple Generation

水面不会凭空振动，需要给水面施加一个振动波源。
论文里面没有提如何产生波源，不同初始化波源的方式得到的波形也不一样。常见的一种方法是圆形初始化。

Without external forces, the water surface would stay calm. In order to have water ripples, vibration sources must be added to the water. The style of the water ripple is different with different ways of initialization. The most common way is to initialize a circle area to a specific value.

引用引用运行体的水波生成与扩散模拟算法的话：
Simulation Algorithms of Water Wave Caused by Moving Body:

水波初始振荡发生时 ,其原动力是负向的 ,振动向负方向偏移。
振荡发生后 ,振动立即会向水面的各个方向传递 ,
其传播到的水域内各点的水纹高度应为该点在上一时刻的振动状态与其相邻点在更前一时刻的原振动状态的综合效应。

According to wave theory, when the oscillation of water ripples happens, 
the vibration would move along the negative z-direction and this vibration would spread to other directions in xy surface 
which means a negative impulse should be added to the mesh to cause the vibration.

对以 $（x,y)$ 为中心 $r$ 为半径的区域范围 $z = A(x,y) = n$ 进行初始化（施加一个负的尖脉冲）实际上正的也没什么区别。。
$r=1$ 时就是一个点，模拟雨点效果不错。简化版：

$A(x,y)$ is a round area with the center $(x,y)$ and the radius $r$ . And use $A(x,y)$ to initialize the mesh(Adding a negative impulse to the mesh, a positive one has no differences).
If $r=1$ , then we just change one value of the node in the mesh. It is not bad to simulate the raindrop effect.

void init(int x, int y, float energy)
{
     nwater2[i][j] = energy;
}

2 水波演变公式推导 Water Ripple Propagation

文章给出了水波演变公式的推导过程，推导过程不是很复杂。
这里简单介绍一下。
波浪力学里面用小波变换来描述施加扰动后自由面的高度：

The article has the derivation process of the water ripple and it is not difficult.
In wave mechanics, small amplitude wave is a common and basic model to describe water wave. According to the small amplitude wave theory, the water height can be described as follow:

$\eta = -\frac{\omega}{g}D\cdot coth(kh)\cdot sin(\omega t+k_{1}x+k_{2}z+\phi)$

本质上是一个带一堆系数的sin函数，参数意义在这里不是很重要。
Generally, it is a complex sine function where parameters are not important here.

令
Let
$\hat t = \frac{t_i+t_{i+1}}{2}$ , $\Delta t = t_{i+1} -t_i (i\ge0)$

利用三角函数和差化积公式可以推出
This function can be deduced:

$\eta(x,z,t_i) + \eta(x,z,t_{i+1}) = 2\eta(x,z,\hat t) cos\frac{\omega\Delta t}{2}$

由于水波自身特点，就是每个点的高度都受周围的点的高度影响。理想的波是各向同性传播的，假设这里的水波也是理想情况。
那么这个 $\eta(x,z,\hat t)$ 可以用周围的点的高度来估计。
假设网格长这样：

Ideally, water wave propagation is isotropic. And the height of a cell is affected by the heights of its adjacent cells in the previous moment.
Then $\eta(x,z,\hat t)$ can be estimated by the heights of its neighbor points.
Suppose the mesh is like this:

mesh03.png

$\eta(x,z,\hat t)= \frac{1}{r}\sum\limits_{j=1}^{r}\eta(x_j,z_j,t_i)$
这里 $r$ 是4，那么令 $2cos\frac{\omega\Delta t}{2} = 1$ ，这里需要注意能量守恒。如果 $r$ 是8，需令 $cos\frac{\omega\Delta t}{2} = 1$ 。论文中只提了 $r = 8$ 的情况。

Here, $r$ is 4，let $2cos\frac{\omega\Delta t}{2} = 1$ , and the law of energy conservation should be taken into consideration. If $r$ is 8，then $cos\frac{\omega\Delta t}{2} = 1$ . The article only mentions when $r = 8$ .

所以演变公式可以变为
Then the evolving formula is
$\eta(x_k,z_k,t_{i+1}) = \frac{1}{4}\sum\limits_{j=1}^{4}\eta(x_j,z_j,t_i)- \eta(x_k,z_k,t_i)$

3 水波消散 Fading Process

公式理想情况下能量符合能量守恒定律的，但现实情况能量会减小但不能增加。如果能量增加，水面效果会爆炸。为了使水波消散，我们对得到的 $\eta$ 乘上一个绝对值小于1的系数即可。
Ideally, it has to follow the law of energy conservation. But in reality, the energy can have the loss. It the energy increased, then the effect of the water surface would be a mess. We just to multiple a variable which has a value between 0 and 1 to make the ripple fade away.

void spread()
{
    for(int i = 1;i < SIZE_WATER-1; i++)
    {
        for(int j = 1; j < SIZE_WATER-1; j++)
        {
            float y = nwater1[i-1][j] + nwater1[i+1][j] + nwater1[i][j-1] + nwater1[i][j+1];
            nwater2[i][j] = y * 0.5f - nwater2[i][j];
            nwater2[i][j] -= nwater2[i][j] / 32.0f;
        }
    }
    for(int i = 1;i < SIZE_WATER-1; i++)
    {
        for(int j = 1; j < SIZE_WATER-1; j++)
        {
            ntmp[i][j] = nwater1[i][j];
            nwater1[i][j] = nwater2[i][j];
            nwater2[i][j] = ntmp[i][j];
        }
    }
}

水波动画效果如下：
The water ripple effect:

mesh04.png

mesh05.png

mesh06.png

mesh07.png

mesh08.png

mesh09.png

mesh10.png

这个算法厉害之处在于他这个简单规则可以有效的处理多个波互相干扰和障碍物反射。增加了能量耗散系数对水波的消退也能很好的模拟。
This algorithm is very awesome that its simple rules can easily handle the interaction of many waves and the obstacles. Adding a loss can simulate the fading process well.

2 水波算法 Water Ripple Simulation

1 波源产生 Water Ripple Generation

2 水波演变公式推导 Water Ripple Propagation

3 水波消散 Fading Process

推荐阅读更多精彩内容