Waves in space-dependent and in time-dependent materials obey similar wave equations, with interchanged time- and space-coordinates. However, since the causality conditions are the same in both types of material (i.e., without interchangement of time- and space-coordinates), the solutions are dissimilar. We present a systematic treatment of wave propagation and scattering in 1D space-dependent and in 1D time-dependent materials. After formulating unified equations, we discuss Green's functions and simple wave field representations for both types of material. Next we discuss propagation invariants, i.e., quantities that are independent of the space coordinate in a space-dependent material (such as the net power-flux density) or of the time coordinate in a time-dependent material (such as the net field-momentum density). A discussion of general reciprocity theorems leads to the well-known source–receiver reciprocity relation for the Green's function of a space-dependent material and a new source–receiver reciprocity relation for the Green's function of a time-dependent material. A discussion of general wave field representations leads to the well-known expression for Green's function retrieval from the correlation of passive measurements in a space-dependent material and a new expression for Green's function retrieval in a time-dependent material. After an introduction of a matrix–vector wave equation, we discuss propagator matrices for both types of material. Since the initial condition for a propagator matrix in a time-dependent material follows from the boundary condition for a propagator matrix in a space-dependent material by interchanging the time- and space-coordinates, the propagator matrices for both types of material are interrelated in the same way. This also applies to representations and reciprocity theorems involving propagator matrices, and to Marchenko-type focusing functions.

Perfectly diffuse wave fields are the underlying assumption for noise-correlation tomography in seismology, nondestructive testing, and elastography; however, perfectly diffuse fields are rarely encountered in real-world applications. We show that homogeneously distributed magnetic microparticles allow instantaneous generation of a diffuse wave field, which can be imaged using a clinical probe connected to a fully programmable ultrasound scanner. The particles are placed inside a bilayered hydrogel and act as elastic-wave sources on excitation by a magnetic pulse. Using ultrafast ultrasound imaging coupled to phase tracking, the diffuse elastic wave field is imaged. This allows the local wave velocity to be measured everywhere on the image using noise-correlation algorithms inspired by seismology. Thanks to this instantaneous diffuse wave field, a very short acquisition time is sufficient to retrieve the wave speed contrast of a bilayered phantom. The correlation time window can be shrunk down to three time samples, which we show in a numerical simulation mimicking the experimental conditions. Our experimental and numerical results are consistent with theoretical predictions made by information theory, and they pave the way for real-time elasticity imaging. This is of particular interest for monitoring of medical treatments through real-time tissue-elasticity assessment, and it is also applicable in related fields such as seismology and nondestructive testing.

Arteriosclerosis is a major risk factor for cardiovascular disease and results in arterial vessel stiffening. Velocity estimation of the pulse wave sent by the heart and propagating into the arteries is a widely accepted biomarker. This symmetrical pulse wave propagates at a speed which is related to the Young's modulus through the Moens Korteweg (MK) equation. Recently, an antisymmetric flexural wave has been observed in vivo. Unlike the symmetrical wave, it is highly dispersive. This property offers promising applications for monitoring arterial stiffness and early detection of atheromatous plaque. However, as far as it is known, no equivalent of the MK equation exists for flexural pulse waves. To bridge this gap, a beam based theory was developed, and approximate analytical solutions were reached. An experiment in soft polymer artery phantoms was built to observe the dispersion of flexural waves. A good agreement was found between the analytical expression derived from beam theory and experiments. Moreover, numerical simulations validated wave speed dependence on the elastic and geometric parameters at low frequencies. Clinical applications, such as arterial age estimation and arterial pressure measurement, are foreseen.