Blog Entry @ Thursday, October 15, 2026, by James Pate Williams, Jr. Solution of the Telegraph Equation

// TelegraphEquation.cpp
// Copyright (c) Wednesday, October 14, 2026
// by James Pate Williams, Jr., BA, BS, MSwE, PhD

#include <iomanip>
#include <iostream>
#include <vector>

static double omega(
    double alpha, double beta, double c, double l, int n)
{
    double pi = 4.0 * atan(1.0);
    double term1 = pow(n * pi * c / l, 2.0);
    double term2 = 0.25 * pow(alpha - beta, 2.0);
    return sqrt(term1 - term2);
}

static double phi(
    double alpha, double beta, double c, double l, int n)
{
    double d = 0.5 * (alpha + beta);
    return atan(d / omega(alpha, beta, c, l, n));
}

static double An(
    double alpha, double beta, double a, double c, double l, int n)
{
    double factor1 = 2.0 / (l, cos(phi(alpha, beta, c, l, n)));
    double factor2 = sin(n * 4.0 * atan(1.0) * a / l);
    return factor1 * factor2;
}

static double analytic_u(
    double alpha, double beta, double a,
    double c, double l, double x, double t,
    int N)
{
    double sum = 0.0;

    for (int n = 1; n <= N; n++)
    {
        double d = 0.5 * (alpha + beta);
        double an = An(alpha, beta, a, c, l, n);
        double factor1 = an * exp(-d * t);
        double omegaN = omega(alpha, beta, c, l, n);
        double phiN = phi(alpha, beta, c, l, n);
        double factor2 = cos(omegaN * t - phiN);
        double factor3 = sin(n * 4.0 * atan(1.0) * x / l);
        sum += factor1 * factor2 * factor3;
    }

    return sum;
}

int main()
{
    char line[128] = "";
    double alpha = 0.0, beta = 0.0, a = 0.0;
    double c = 0.0, l = 0.0;
    int N = 0, nPoints = 0;
    std::cout << "alpha   = ";
    std::cin.getline(line, 128);
    alpha = atof(line);
    std::cout << "beta    = ";
    std::cin.getline(line, 128);
    beta = atof(line);
    std::cout << "a       = ";
    std::cin.getline(line, 128);
    a = atof(line);
    std::cout << "c       = ";
    std::cin.getline(line, 128);
    c = atof(line);
    std::cout << "l       = ";
    std::cin.getline(line, 128);
    l = atof(line);
    std::cout << "N       = ";
    std::cin.getline(line, 128);
    N = atoi(line);
    std::cout << "nPoints = ";
    std::cin.getline(line, 128);
    nPoints = atoi(line);
    double deltaX = 1.0 / nPoints;
    double deltaT = 1.0 / nPoints;
    std::vector<double> u(nPoints), x(nPoints), t(nPoints);

    for (int i = 0; i < nPoints; i++)
    {
        x[i] = i * deltaX;
        t[i] = i * deltaT;
        u[i] = analytic_u(alpha, beta, a, c, l, x[i], t[i], N);
        std::cout << std::setprecision(10);
        std::cout << std::fixed;
        std::cout << x[i] << '\t' << t[i] << '\t';
        
        if (u[i] >= 0.0)
            std::cout << ' ';

        std::cout << u[i] << std::endl;
    }

    return 0;
}