To sample the wave equation for string vibration in real time, you would typically use a technique called digital sampling. Here's a general overview of the process:
Discretize the string: Divide the string into a series of discrete points or nodes. Each node represents a specific position along the string.
Define the initial conditions: Assign initial displacement and velocity values to each node on the string. These initial conditions could be based on the desired waveform or any specific excitation applied to the string.
Set the sampling rate: Determine the sampling rate at which you want to capture the string's vibration. The sampling rate determines how many samples per second will be taken.
Update the string's state: In each time step, calculate the displacement of each node based on the wave equation and the values of neighboring nodes. The wave equation considers factors such as tension, mass per unit length, and the second derivative of displacement with respect to position and time.
Record the samples: At each time step, sample the displacement of specific nodes on the string. The sampling rate determines how frequently you capture the samples. You can store these samples in an array or buffer for further processing or real-time playback.
Update the time: Increment the time by a small step, determined by the sampling rate, and repeat steps 4 and 5 to calculate and record the next set of samples.
By repeating these steps, you can simulate the real-time behavior of the string vibration and capture the corresponding samples. These samples can then be used for various purposes such as visualization, analysis, or synthesis of the string's sound in real time.
It's worth noting that the specific implementation details can vary depending on the software or programming language you're using, as well as the numerical methods employed for solving the wave equation.