Global AI and Data Science

Good day, awesome IBM communty,

The journey from understanding elastic wave propagation to complex quantum systems is both insightful and transformative. In this blog post, we explore the 1D elastic wave equation and its numerical solution using the Finite Difference Method (FDM). Grasping the 1D elastic wave equation and the FDM provides a nice foundation before diving into more complex spaces, such as quantum or complex vector spaces.

Conceptual Building Blocks:

The 1D elastic wave equation models how stress and velocity propagate through an elastic medium, described using partial differential equations (PDEs). These PDEs are essential in classical mechanics, electromagnetism, and quantum mechanics. By studying them, we build intuition for:

• Field propagation (e.g., stress and velocity fields in the case of elastic waves, wavefunctions in quantum mechanics).

• Discretization of continuous problems, such as turning PDEs into numerical problems that can be solved on a grid.

• Finite difference approximations give us insight into how to numerically solve differential equations, which is a core-step when solving Schrödinger's equation in quantum mechanics.

'stress refers to the internal force per unit area that is exerted within a material due to an external force acting on it.'

 Let's understand this with an analogy. For a rubber band:

• Tension (positive stress): When you pull on the rubber band, it stretches. The atoms inside the rubber band feel a force that is pulling them apart. This force that stretches or pulls the material is called tension, and it's one type of stress.
• Compression (negative stress): If you push on a rubber object (like squishing a soft ball), the atoms inside get squeezed together. This squishing force is called compression, and it's another type of stress.

Stress in the 1D Elastic Wave Equation

In the 1D elastic wave equation, stress $\sigma (x,t)$ is one of the two primary variables (the other being velocity $v(x,t)$) that describe how elastic waves travel through the medium. The two equations that describe how stress and velocity evolve over time are:

1. Stress equation:

1. • This equation tells us how the rate of change of stress over time depends on the spatial rate of change of velocity (how the velocity varies across the material).
   • Is the shear modulus of the material, a property that describes the material's resistance to deformation.

2. Velocity equation:

   
   
   • This equation describes how the rate of change of velocity over time depends on how stress changes in space (i.e., how unevenly stress is distributed along the material).
   •  Is the density of the medium, and is the external force applied to the system.

These equations relate stress, velocity, density, and shear modulus, which are analogous to quantities in quantum mechanics (e.g., potential, probability density, wavefunctions).

Wave 1 and Wave 2 are two sine waves propagating in opposite directions.

• The function wave1 describes a wave moving to the right, while wave2 describes a wave moving to the left.
• The superposition is calculated by simply adding the two waves at each time step. This represents the core principle of wave superposition, where multiple waves overlap to form a more complex wave pattern.
• Interpret how two waves move and interfere with each other, generating constructive or destructive interference at different points in the space.

Wave Propagation Simulation:

• The code which simulates wave propagation of the above plots using the wave equation, can be solved numerically using finite difference methods. The grid-staggering concept and finite difference approximations are crucial in numerical simulations of wave propagation, and they directly related with the previous figure.

In the finite difference method, the physical domain is discretized on a grid. In staggered grids, the variables $v$ and $\sigma$ are defined at different spatial points:

• Velocity is defined at points (integer indices).
• Stress is defined at midpoints

This allows for better stability and accuracy in solving wave propagation problems. Grid-staggering helps in computing the derivatives without losing precision by averaging the positions of the stress and velocity.

For the staggered grid, the discrete versions of the equations become:

​​

Where Δt and Δx are the time step and space step, respectively.

For numerical solutions, partial derivatives are replaced by finite difference approximations. Specifically:

• Time derivatives are discretized using central difference approximations, for example:

  ​

• Space derivatives are also approximated as central differences:

  ​

Similarly, the stress evolution equation is discretized as:

​

and

These centered differences are known to be second-order accurate.

The scheme involves solving for velocity and stress in an alternating fashion using the discretized equations. The velocity is updated first based on the stress, then the stress is updated using the velocity.

The velocity is updated using the discretized form of the first equation:

This is the extrapolation of velocity at time step j+1/2 based on the stress gradient and the external source ​. The stress is updated using the discretized form of the second equation:

This gives the stress at time  j+1 based on the velocity gradient.

The source term is modeled using a Gaussian derivative function. The source excites the wavefield at the source location . Mathematically, the source can be represented as:

where a=4T₀​, and T₀ is the period related to the central frequency f₀.

The time step Δt is chosen to satisfy the Courant-Friedrichs-Lewy (CFL) stability condition for numerical stability:

where c is the wave velocity. In the code, the stability factor ϵ is used to determine the time step.

In both classical wave mechanics (like elastic waves) and quantum mechanics, the idea of wave propagation is central. The 1D wave equation introduces the concept of superposition of waves, as solutions to linear equations can be added to create more complex wave patterns. This is a precursor to understanding wave-particle duality in quantum mechanics [1], where quantum particles (like electrons) can exhibit both wave and particle properties.

Complex space often involves analyzing behaviors in multi-dimensional systems (whether physical, mathematical, or quantum). By understanding how waves propagate in 1D finite difference models, we gain insight into the mathematics of differential equations governing wave behavior—equations also in quantum mechanics. For instance, Schrödinger's equation, fundamental in quantum physics, is a wave equation. The skills in discretizing and approximating wave equations from classical systems transfer directly when we encounter quantum wave functions.

• Discretization techniques, such as the finite difference approach used in grid-staggering, form the basis for many numerical simulations in quantum physics. Quantum systems are often analyzed using finite grids (like lattice models in quantum field theory or tight-binding approximations in solid-state physics).

Knowing how to break continuous systems into discrete grids (and the trade-offs in precision) can allow us for better modeling in quantum spaces, where quantum waves (or probability distributions) evolve over space and time. Stability is critical when dealing with complex systems or quantum algorithms, where small inaccuracies can cause major deviations. The Courant-Friedrichs-Lewy (CFL) condition, which is used to maintain stability in numerical solutions, plays a similar role when ensuring the convergence of quantum algorithms or simulations. Quantum systems require precise control over time-evolution and space to preserve quantum coherence.

Thank you for taking the time to read this blog post! I hope you found the exploration of elastic wave propagation and its connections insightful. Any feedback or suggestions are always welcome—it's a pleasure to share and learn together. Best regards.